Class 12th Chapter -10 Vector Algebra | NCERT Maths Solution | NCERT Solution

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NCERT Solutions for Class 12 Maths Chapter :10 Vector Algebra

Class 12 Maths Chapter 10 Vector Algebra Ex 10.1

Ex 10.1 Class 12 Maths Question 1.
Represent graphically a displacement of 40km, 30° east of north.
Solution:
A line segment of 2 cm is drawn on the right of OY making an angle of 30° with it. OP = 40 km,
scale 1cm = 20 km. Vector $\overrightarrow { OP }$

represents displacement of 40 km 30° east of north.

NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.1 Q1.1

Ex 10.1 Class 12 Maths Question 2.
Classify the following measures as scalars and vectors.
(i) 10 kg
(ii) 2 metres north- west
(iii) 40°
(iv) 40 watt
(v) 10^-19coulomb
(vi) 20 m/sec².
Solution:
(i) Mass-scalar
(ii) Directed distance-vector
(iii) Temperature-scalar
(iv) Rate of electricity-scalar
(v) Electric charge-vector
(vi) Acceleration-vector

Ex 10.1 Class 12 Maths Question 3.
Classify the following as scalar and vector quantities
(i) time period
(ii) distance
(iii) force
(iv) velocity
(v)work.
Solution:
Scalar Quantity: (i) time period (ii) distance (v) work.
Vector Quantity: (iii) force (iv) velocity

Ex 10.1 Class 12 Maths Question 4.
In a square, identify the following vectors
(i) Co-initial
(ii) Equal
(iii) collinear but not equal
Solution:
(i) Co initial vectors are $\overrightarrow { a } ,\overrightarrow { d }$
(ii) Equal Vectors are $\overrightarrow { b } ,\overrightarrow { d }$
(iii) Collinear but not equal vectors are $\overrightarrow { a } ,\overrightarrow { c }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.1 Q4.1

Ex 10.1 Class 12 Maths Question 5.
Answer the following as true or false:
(i) $\overrightarrow { a } ,\overrightarrow { -a }$ are collinear.
(ii) Two collinear vectors are always equal in magnitude.
(iii) Two vectors having same magnitude are collinear.
(iv) Two collinear vectors having the same magnitude are equal.
Solution:
(i) True
(ii) False
(iii) False
(iv) False.

Class 12 Maths Chapter 10 Vector Algebra Ex 10.2

Ex 10.2 Class 12 Maths Question 1.
Compute the magnitude of the following vectors:
$\overrightarrow { a } =\hat { i } +\hat { j } +\hat { k } ,\overrightarrow { b } =\hat { 2i } -\hat { 7j } -\hat { 3k }$
$\overrightarrow { c } =\frac { 1 }{ \sqrt { 3 } } \hat { i } +\frac { 1 }{ \sqrt { 3 } } \hat { j } -\frac { 1 }{ \sqrt { 3 } } \hat { k }$
Solution:
$\overrightarrow { a } =\hat { i } +\hat { j } +\hat { k }$
$\left| \overrightarrow { a } \right| =\sqrt { { 1 }^{ 2 }+{ 1 }^{ 2 }+{ 1 }^{ 2 } }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.2 Q1.1

Ex 10.2 Class 12 Maths Question 2.
Write two different vectors having same magnitude.
Solution:
$\overrightarrow { a } =\hat { i } +\hat { 2j } +\hat { 3k } ,\overrightarrow { b } =\hat { 3i } +\hat { 2j } +\hat { k }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.2 Q2.1
Such possible answers are infinite

Ex 10.2 Class 12 Maths Question 3.
Write two different vectors having same direction.
Solution:
Let the two vectors be
$\overrightarrow { a } =\hat { i } +\hat { j } +\hat { k } ,\overrightarrow { b } =\hat { 3i } +\hat { 3j } +\hat { 3k }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.2 Q3.1
Hence vectors $\overrightarrow { a } ,\overrightarrow { b }$ have the same direction but different magnitude

Ex 10.2 Class 12 Maths Question 4.
Find the values of x and y so that the vectors $\overrightarrow { 2i } +\overrightarrow { 3j } \quad and\quad \hat { xi } +\hat { yj }$ are equal.
Solution:
We are given $\overrightarrow { 2i } +\overrightarrow { 3j } \quad and\quad \hat { xi } +\hat { yj }$
If vectors are equal, then their respective components are equal. Hence x = 2, y = 3.

Ex 10.2 Class 12 Maths Question 5.
Find the scalar and vector components of the vector with initial point (2,1) and terminal point (-5,7).
Solution:
LetA(2, 1) be the initial point and B(-5,7) be the terminal point $\overrightarrow { AB } =\left( { x }_{ 2 }-{ x }_{ 1 } \right) \hat { i } +\left( { y }_{ 2 }-{ y }_{ 1 } \right) \hat { j } =-\hat { 7i } +\hat { 6j }$
∴The vector components are $-\hat { 7i } and\hat { 6j }$ and scalar components are – 7 and 6.

Ex 10.2 Class 12 Maths Question 6.
Find the sum of three vectors:
$\overrightarrow { a } =\hat { i } -\hat { 2j } +\hat { k } ,\overrightarrow { b } =-2\hat { i } +\hat { 4j } +5\hat { k } \quad and\quad \overrightarrow { c } =\hat { i } -\hat { 6j } -\hat { 7k } ,$
Solution:
$\overrightarrow { a } =\hat { i } -\hat { 2j } +\hat { k } ,\overrightarrow { b } =-2\hat { i } +\hat { 4j } +5\hat { k } \quad and\quad \overrightarrow { c } =\hat { i } -\hat { 6j } -\hat { 7k } ,$
$\overrightarrow { a } +\overrightarrow { b } +\overrightarrow { c } =\hat { 0i } -\hat { 4j } -\hat { k } =-4\hat { i } -\hat { k }$

Ex 10.2 Class 12 Maths Question 7.
Find the unit vector in the direction of the vector
$\overrightarrow { a } =\hat { i } +\hat { j } +\hat { 2k }$
Solution:
$\overrightarrow { a } =\hat { i } +\hat { j } +\hat { 2k }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.2 Q18.2

Ex 10.2 Class 12 Maths Question 8.
Find the unit vector in the direction of vector $\overrightarrow { PQ }$ , where P and Q are the points (1,2,3) and (4,5,6) respectively.
Solution:
The points P and Q are (1, 2, 3) and (4, 5, 6) respectively
$\overrightarrow { PQ } =(4-1)\hat { i } +(5-2)\hat { j } +(6-3)\hat { k }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.2 Q8.1

Ex 10.2 Class 12 Maths Question 9.
For given vectors $\overrightarrow { a } =2\hat { i } -\hat { j } +2\hat { k } \quad and\quad \overrightarrow { b } =-\hat { i } +\hat { j } -\hat { k }$ find the unit vector in the direction of the vector $\overrightarrow { a } +\overrightarrow { b }$
Solution:
$\overrightarrow { a } =2\hat { i } -\hat { j } +2\hat { k } \quad and\quad \overrightarrow { b } =-\hat { i } +\hat { j } -\hat { k }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.2 Q9.1

Ex 10.2 Class 12 Maths Question 10.
Find a vector in the direction of $5\hat { i } -\hat { j } +2\hat { k }$ which has magnitude 8 units.
Solution:
The given vector is $\overrightarrow { a } =5\hat { i } -\hat { j } +2\hat { k }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.2 Q10.1

Ex 10.2 Class 12 Maths Question 11.
Show that the vector $2\hat { i } -3\hat { j } +4\hat { k } \quad and\quad -4\hat { i } +6\hat { j } -8\hat { k }$ are collinear.
Solution:
$\overrightarrow { a } =2\hat { i } -3\hat { j } +4\hat { k } \quad and\quad \overrightarrow { b } =-4\hat { i } +6\hat { j } -8\hat { k }$
$=-2(2\hat { i } -3\hat { j } +4\hat { k } )$
vector $\overrightarrow { a } \quad and\quad \overrightarrow { b }$ have the same direction they are collinear.

Ex 10.2 Class 12 Maths Question 12.
Find the direction cosines of the vector $\hat { i } +2\hat { j } +3\hat { k }$
Solution:
let $\overrightarrow { p } =\hat { i } +2\hat { j } +3\hat { k }$
Now a = 1,b = 2,c = 3
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.2 Q12.1

Ex 10.2 Class 12 Maths Question 13.
Find the direction cosines of the vector joining the points A (1,2, -3) and B(-1, -2,1), directed fromAtoB.
Solution:
Vector joining the points A and B is
$({ x }_{ 2 }-{ x }_{ 1 })\hat { i } +({ y }_{ 2 }-{ y }_{ 1 })\hat { j } +({ z }_{ 2 }-{ z }_{ 1 })\hat { k }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.2 Q13.1

Ex 10.2 Class 12 Maths Question 14.
Show that the vector $\hat { i } +\hat { j } +\hat { k }$ are equally inclined to the axes OX, OY, OZ.
Solution:
Let $\hat { i } +\hat { j } +\hat { k } =\overrightarrow { a }$ , Direction cosines of vector $x\hat { i } +y\hat { j } +z\hat { k }$ are
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.2 Q14.1
which shows that the vector a is equally inclined to the axes OX, OY, OZ.

Ex 10.2 Class 12 Maths Question 15.
Find the position vector of a point R which divides the line joining the points whose positive vector are $P(\hat { i } +2\hat { j } -\hat { k } )\quad and\quad Q(-\hat { i } +\hat { j } +\hat { k } )$ in the ratio 2:1
(i) internally
(ii) externally.
Solution:
(i) The point R which divides the line joining the point $P(\overrightarrow { a } )\quad and\quad Q(\overrightarrow { b } )$ in the ratio m : n
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.2 Q15.1

Ex 10.2 Class 12 Maths Question 16.
Find position vector of the mid point of the vector joining the points P (2,3,4) and Q (4,1, -2).
Solution:
Let $\overrightarrow { OP } =2\hat { i } +3\hat { j } +4\hat { k } \quad and\quad \overrightarrow { OQ } =4\hat { i } +\hat { j } -2\hat { k }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.2 Q16.1

Ex 10.2 Class 12 Maths Question 17.
Show that the points A, B and C with position vector $\overrightarrow { a } =3\hat { i } -4\hat { j } -4\hat { k } ,\overrightarrow { b } =2\hat { i } -\hat { j } +\hat { k } and\quad \overrightarrow { c } =\hat { i } -3\hat { j } -5\hat { k }$ respectively form the vertices of a right angled triangle.
Solution:
$\overrightarrow { AB } =\overrightarrow { b } -\overrightarrow { a } =-\hat { i } +3\hat { j } +5\hat { k }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.2 Q17.1

Ex 10.2 Class 12 Maths Question 18.
In triangle ABC (fig.), which of the following is not
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.2 Q18.1
(a) $\overrightarrow { AB } +\overrightarrow { BC } +\overrightarrow { CA } =\overrightarrow { 0 }$
(b) $\overrightarrow { AB } +\overrightarrow { BC } -\overrightarrow { AC } =\overrightarrow { 0 }$
(c) $\overrightarrow { AB } +\overrightarrow { BC } -\overrightarrow { CA } =\overrightarrow { 0 }$
(d) $\overrightarrow { AB } -\overrightarrow { CB } +\overrightarrow { CA } =\overrightarrow { 0 }$
Solution:
We know that
$\overrightarrow { AB } +\overrightarrow { BC } +\overrightarrow { CA } =\overrightarrow { 0 }$
$\overrightarrow { AB } +\overrightarrow { BC } -\overrightarrow { AC } =\overrightarrow { 0 }$
Hence option (c) is not correct

Ex 10.2 Class 12 Maths Question 19.
If $\overrightarrow { a } ,\overrightarrow { b }$ are two collinear vectors then which of the following are incorrect:
(a) $\overrightarrow { b } =\lambda \overrightarrow { a }$ , for some scalar λ.
(b) $\overrightarrow { a } =\pm \overrightarrow { b }$
(c) the respective components of $\overrightarrow { a } ,\overrightarrow { b }$ are proportional.
(d) both the vectors $\overrightarrow { a } ,\overrightarrow { b }$ have same direction, but different magnitudes.
Solution:
Options (d) is incorrect since both the vectors $\overrightarrow { a } ,\overrightarrow { b }$ , being collinear, are not necessarily in the same direction. They may have opposite directions. Their magnitudes may be different.

Class 12 Maths Chapter 10 Vector Algebra Ex 10.3

Ex 10.3 Class 12 Maths Question 1.
Find the angle between two vectors $\overrightarrow { a } ,\overrightarrow { b }$ with magnitudes √3 and 2 respectively, and such that $\overrightarrow { a } \cdot \overrightarrow { b } =\sqrt { 6 }$
Solution:
Angle θ between two vectors $\overrightarrow { a } ,\overrightarrow { b }$ ,
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q1.1

Ex 10.3 Class 12 Maths Question 2.
Find the angle between the vectors $\hat { i } -2\hat { j } +3\hat { k } \quad and\quad 3\hat { i } -2\hat { j } +\hat { k }$
Solution:
Let $\overrightarrow { a } =\hat { i } -2\hat { j } +3\hat { k } \quad and\quad \overrightarrow { b } =3\hat { i } -2\hat { j } +\hat { k }$
Let θ be the angle between $\overrightarrow { a } ,\overrightarrow { b }$ ,
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q2.1

Ex 10.3 Class 12 Maths Question 3.
Find the projection of the vector $\overrightarrow { i } -\overrightarrow { j }$ , on the line represented by the vector $\overrightarrow { i } +\overrightarrow { j }$ ,
Solution:
let $\overrightarrow { a } =\hat { i } -\hat { j } \quad and\quad \overrightarrow { b } =\hat { i } +\hat { j }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q3.1

Ex 10.3 Class 12 Maths Question 4.
Find the projection of the vector $\hat { i } +3\hat { j } +7\hat { k }$ on the vector $7\hat { i } -\hat { j } +8\hat { k }$
Solution:
let $\overrightarrow { a } =\hat { i } +3\hat { j } +7\hat { k } \quad and\quad \overrightarrow { b } =7\hat { i } -\hat { j } +8\hat { k }$ then
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q4.1

Ex 10.3 Class 12 Maths Question 5.
Show that each of the given three vectors is a unit vector $\frac { 1 }{ 7 } \left( 2\hat { i } +3\hat { j } +6\hat { k } \right) ,\frac { 1 }{ 7 } \left( 3\hat { i } -6\hat { j } +2\hat { k } \right) ,\frac { 1 }{ 7 } \left( 6\hat { i } +2\hat { j } -3\hat { k } \right)$ Also show that they are mutually perpendicular to each other.
Solution:
$Let\quad \overrightarrow { a } =\frac { 1 }{ 7 } \left( 2\hat { i } +3\hat { j } +6\hat { k } \right) ,\overrightarrow { b } =\frac { 1 }{ 7 } \left( 3\hat { i } -6\hat { j } +2\hat { k } \right) ,\overrightarrow { c } =\frac { 1 }{ 7 } \left( 6\hat { i } +2\hat { j } -3\hat { k } \right)$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q5.1

Ex 10.3 Class 12 Maths Question 6.
$Find\left| \overrightarrow { a } \right| and\left| \overrightarrow { b } \right| if\left( \overrightarrow { a } +\overrightarrow { b } \right) \cdot \left( \overrightarrow { a } -\overrightarrow { b } \right) =8\quad and\left| \overrightarrow { a } \right| =8\left| \overrightarrow { b } \right|$
Solution:
Given $\left( \overrightarrow { a } +\overrightarrow { b } \right) \cdot \left( \overrightarrow { a } -\overrightarrow { b } \right) =8$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q6.1

Ex 10.3 Class 12 Maths Question 7.
Evaluate the product :
$\left( 3\overrightarrow { a } -5\overrightarrow { b } \right) \cdot \left( 2\overrightarrow { a } +7\overrightarrow { b } \right)$
Solution:
$\left( 3\overrightarrow { a } -5\overrightarrow { b } \right) \cdot \left( 2\overrightarrow { a } +7\overrightarrow { b } \right)$
$=6\overrightarrow { a } .\overrightarrow { a } -10\overrightarrow { b } \overrightarrow { a } +21\overrightarrow { a } .\overrightarrow { b } -35\overrightarrow { b } .\overrightarrow { b }$
$=6{ \left| \overrightarrow { a } \right| }^{ 2 }-11\overrightarrow { a } \overrightarrow { b } -35{ \left| \overrightarrow { b } \right| }^{ 2 }$

Ex 10.3 Class 12 Maths Question 8.
Find the magnitude of two vectors $\overrightarrow { a } ,\overrightarrow { b }$ having the same magnitude and such that the angle between them is 60° and their scalar product is $\frac { 1 }{ 2 }$
Solution:
We know that $\overrightarrow { a } .\overrightarrow { b } =\left| \overrightarrow { a } \right| \left| \overrightarrow { b } \right| cos\theta$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q8.1

Ex 10.3 Class 12 Maths Question 9.
Find $\left| \overrightarrow { x } \right|$ , if for a unit vector $\overrightarrow { a } ,(\overrightarrow { x } -\overrightarrow { a } )\cdot (\overrightarrow { x } +\overrightarrow { a } )=12$
Solution:
Given
$\overrightarrow { a } ,(\overrightarrow { x } -\overrightarrow { a } )\cdot (\overrightarrow { x } +\overrightarrow { a } )=12$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q9.1

Ex 10.3 Class 12 Maths Question 10.
If $\overrightarrow { a } =2\hat { i } +2\hat { j } +3\hat { k } ,\overrightarrow { b } =-\hat { i } +2\hat { j } +\hat { k } and\overrightarrow { c } =3\hat { i } +\hat { j }$ such that $\overrightarrow { a } +\lambda \overrightarrow { b } \bot \overrightarrow { c }$ , then find the value of λ.
Solution:
Given
$\overrightarrow { a } =2\hat { i } +2\hat { j } +3\hat { k } ,\overrightarrow { b } =-\hat { i } +2\hat { j } +\hat { k } and\overrightarrow { c } =3\hat { i } +\hat { j }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q10.1

Ex 10.3 Class 12 Maths Question 11.
Show that $\left| \overrightarrow { a } \right| \overrightarrow { b } +\left| \overrightarrow { b } \right| a\quad \bot \quad \left| \overrightarrow { a } \right| \cdot \overrightarrow { b } -\left| \overrightarrow { b } \right| a$ for any two non-zero vectors $\overrightarrow { a } ,\overrightarrow { b }$
Solution:
$\overrightarrow { a } ,\overrightarrow { b }$ are any two non zero vectors
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q11.1

Ex 10.3 Class 12 Maths Question 12.
If $\overrightarrow { a } \cdot \overrightarrow { a } =0\quad and\quad \overrightarrow { a } \cdot \overrightarrow { b } =0$ , then what can be concluded about the vector $\overrightarrow { b }$ ?
Solution:
$\overrightarrow { a } \overrightarrow { a } =0\quad and\quad \overrightarrow { a } .\overrightarrow { b } =0$ ,
=> $\overrightarrow { b }$ = 0
Hence b is any vector.

Ex 10.3 Class 12 Maths Question 13.
If $\overrightarrow { a } ,\overrightarrow { b } ,\overrightarrow { c }$ are the unit vector such that $\overrightarrow { a } +\overrightarrow { b } +\overrightarrow { c } =0$ , then find the value of $\overrightarrow { a } .\overrightarrow { b } +\overrightarrow { b } .\overrightarrow { c } +\overrightarrow { c } .\overrightarrow { a }$
Solution:
We have
$\overrightarrow { a } +\overrightarrow { b } +\overrightarrow { c } =0$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q13.1

Ex 10.3 Class 12 Maths Question 14.
If either vector $\overrightarrow { a } =0\quad or\quad \overrightarrow { b } =0$ then $\overrightarrow { a } .\overrightarrow { b } =0$ . But the converse need not be true. Justify your answer with an example.
Solution:
Given: $\overrightarrow { a } =0\quad or\quad \overrightarrow { b } =0$
To prove: $\overrightarrow { a } .\overrightarrow { b } =0$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q14.1

Ex 10.3 Class 12 Maths Question 15.
If the vertices A,B,C of a triangle ABC are (1,2,3) (-1,0,0), (0,1,2) respectively, then find ∠ABC.
Solution:
Let O be the origin then.
$\frac { 1 }{ 2 }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q15.1

Ex 10.3 Class 12 Maths Question 16.
Show that the points A (1,2,7), B (2,6,3) and C (3,10, -1) are collinear.
Solution:
The position vectors of points A, B, C are
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q16.1

Ex 10.3 Class 12 Maths Question 17.
Show that the vectors $2\hat { i } -\hat { j } +\hat { k } ,\hat { i } -3\hat { j } -5\hat { k }$ and $\left( 3\hat { i } -4\hat { j } -4\hat { k } \right)$ from the vertices of a right angled triangle.
Solution:
The position vectors of the points A, B and C are
$2\hat { i } -\hat { j } +\hat { k } ,\hat { i } -3\hat { j } -5\hat { k }$ and $\left( 3\hat { i } -4\hat { j } -4\hat { k } \right)$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q17.1

Ex 10.3 Class 12 Maths Question 18.
If $\overrightarrow { a }$ is a non-zero vector of magnitude ‘a’ and λ is a non- zero scalar, then λ $\overrightarrow { a }$ is unit vector if
(a) λ = 1
(b) λ = – 1
(c) a = |λ|
(d) a = $\frac { 1 }{ \left| \lambda \right| }$
Solution:
$\left| \overrightarrow { a } \right| =a$
Given : $\lambda \overrightarrow { a }$ is a unit vectors
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q18.1

Class 12 Maths Chapter 10 Vector Algebra Ex 10.4

Ex 10.4 Class 12 Maths Question 1.
Find $\left| \overrightarrow { a } \times \overrightarrow { b } \right| ,if\quad \overrightarrow { a } =\hat { i } -7\hat { j } +7\hat { k } \quad and\quad \overrightarrow { b } =3\hat { i } -2\hat { j } +2\hat { k }$
Solution:
Given
$\overrightarrow { a } =\hat { i } -7\hat { j } +7\hat { k } \quad and\quad \overrightarrow { b } =3\hat { i } -2\hat { j } +2\hat { k }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.4 Q1.1

NCERT Maths Class 12 Chapter 10

Ex 10.4 Class 12 Maths Question 2.
Find a unit vector perpendicular to each of the vector $\overrightarrow { a } +\overrightarrow { b } \quad and\quad \overrightarrow { a } -\overrightarrow { b }$ , where $\overrightarrow { a } =3\hat { i } +2\hat { j } +2\hat { k } \quad and\quad \overrightarrow { b } =\hat { i } +2\hat { j } -2\hat { k }$
Solution:
we have
$\overrightarrow { a } =3\hat { i } +2\hat { j } +2\hat { k } \quad and\quad \overrightarrow { b } =\hat { i } +2\hat { j } -2\hat { k }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.4 Q2.1

Ex 10.4 Class 12 Maths Question 3.
If a unit vector $\overrightarrow { a }$ makes angle $\frac { \pi }{ 3 } with\quad \hat { i } ,\frac { \pi }{ 4 } with\quad \hat { j }$ and an acute angle θ with $\overrightarrow { k }$ ,then find θ and hence the components of $\overrightarrow { a }$ .
Solution:
$Let\quad \overrightarrow { a } ={ a }_{ 1 }\hat { i } +{ a }_{ 2 }\hat { j } +{ a }_{ 3 }\hat { k } such\quad that\quad \left| \overrightarrow { a } \right| =1$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.4 Q3.1

Ex 10.4 Class 12 Maths Question 4.
Show that $\left( \overrightarrow { a } -\overrightarrow { b } \right) \times \left( \overrightarrow { a } +\overrightarrow { b } \right) =2\left( \overrightarrow { a } \times \overrightarrow { b } \right)$
Solution:
LHS = $\left( \overrightarrow { a } -\overrightarrow { b } \right) \times \left( \overrightarrow { a } +\overrightarrow { b } \right)$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.4 Q4.1

Ex 10.4 Class 12 Maths Question 5.
Find λ and μ if
$\left( 2\hat { i } +6\hat { j } +27\hat { k } \right) \times \left( \hat { i } +\lambda \hat { j } +\mu \hat { k } \right) =0$
Solution:
$\left( 2\hat { i } +6\hat { j } +27\hat { k } \right) \times \left( \hat { i } +\lambda \hat { j } +\mu \hat { k } \right) =0$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.4 Q5.1

Ex 10.4 Class 12 Maths Question 6.
Given that $\overrightarrow { a } .\overrightarrow { b } =0\quad and\quad \overrightarrow { a } \times \overrightarrow { b } =0$ . What can you conclude about the vectors $\overrightarrow { a } ,\overrightarrow { b }$ ?
Solution:
$\overrightarrow { a } .\overrightarrow { b } =0\quad and\quad \overrightarrow { a } \times \overrightarrow { b } =0$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.4 Q6.1

Ex 10.4 Class 12 Maths Question 7.
Let the vectors $\overrightarrow { a } ,\overrightarrow { b } ,\overrightarrow { c }$ are given ${ a }_{ 1 }\hat { i } +{ a }_{ 2 }\hat { j } +{ a }_{ 3 }\hat { k } ,{ b }_{ 1 }\hat { i } +{ b }_{ 2 }\hat { j } +{ b }_{ 3 }\hat { k } ,{ c }_{ 1 }\hat { i } +{ c }_{ 2 }\hat { j } +{ c }_{ 3 }\hat { k }$ . Then show that $\overrightarrow { a } \times \left( \overrightarrow { b } +\overrightarrow { c } \right)$ $=\overrightarrow { a } \times \overrightarrow { b } +\overrightarrow { a } \times \overrightarrow { c }$
Solution:
Given
$\overrightarrow { a } ,\overrightarrow { b } ,\overrightarrow { c }$ are given ${ a }_{ 1 }\hat { i } +{ a }_{ 2 }\hat { j } +{ a }_{ 3 }\hat { k } ,{ b }_{ 1 }\hat { i } +{ b }_{ 2 }\hat { j } +{ b }_{ 3 }\hat { k } ,{ c }_{ 1 }\hat { i } +{ c }_{ 2 }\hat { j } +{ c }_{ 3 }\hat { k }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.4 Q7.1

Ex 10.4 Class 12 Maths Question 8.
If either $\overrightarrow { a } =0\quad or\quad \overrightarrow { b } =0\quad then\quad \hat { a } \times \hat { b } =0$ .Is the
converse true? Justify your answer with an example.
Solution:
$\overrightarrow { a } =0\Rightarrow \left| \overrightarrow { a } \right| =0$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.4 Q8.1

Ex 10.4 Class 12 Maths Question 9.
Find the area of the triangle with vertices A (1,1,2), B (2,3,5) and C (1,5,5).
Solution:
A (1,1,2), B (2,3,5) and C (1,5,5).
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 9

Ex 10.4 Class 12 Maths Question 10.
Find the area of the parallelogram whose adjacent sides are determined by the vectors $\overrightarrow { a } =\hat { i } -\hat { j } +3\hat { k } ,\overrightarrow { b } =2\hat { i } -7\hat { j } +\hat { k }$
Solution:
We have $\overrightarrow { a } =\hat { i } -\hat { j } +3\hat { k } ,\overrightarrow { b } =2\hat { i } -7\hat { j } +\hat { k }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.4 Q10.1

Ex 10.4 Class 12 Maths Question 11.
Let the vectors $\overrightarrow { a } ,\overrightarrow { b }$ such that $\left| \overrightarrow { a } \right| =3,\left| \overrightarrow { b } \right| =\frac { \sqrt { 2 } }{ 3 }$ then $\overrightarrow { a } \times \overrightarrow { b }$ is a unit vector if the angle between $\overrightarrow { a } ,\overrightarrow { b }$ is
(a) $\frac { \pi }{ 6 }$
(b) $\frac { \pi }{ 4 }$
(c) $\frac { \pi }{ 3 }$
(d) $\frac { \pi }{ 2 }$
Solution:
Given
$\left| \overrightarrow { a } \times \overrightarrow { b } \right| =1$
$\left| \overrightarrow { a } \right| =3,\left| \overrightarrow { b } \right| =\frac { \sqrt { 2 } }{ 3 }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.4 Q11.1

Ex 10.4 Class 12 Maths Question 12.
Area of a rectangles having vertices
$A\left( -\hat { i } +\frac { 1 }{ 2 } \hat { j } +4\hat { k } \right) ,B\left( \hat { i } +\frac { 1 }{ 2 } \hat { j } +4\hat { k } \right) ,$
$C\left( \hat { i } -\frac { 1 }{ 2 } \hat { j } +4\hat { k } \right) ,D\left( -\hat { i } -\frac { 1 }{ 2 } \hat { j } +4\hat { k } \right) ,$
(a) $\frac { 1 }{ 2 }$ sq units
(b) 1sq.units
(c) 2sq.units
(d) 4sq.units
Solution:
$\overrightarrow { OA } =\left( -\hat { i } +\frac { 1 }{ 2 } \hat { j } +4\hat { k } \right)$
$\overrightarrow { OB } =\left( \hat { i } +\frac { 1 }{ 2 } \hat { j } +4\hat { k } \right)$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.4 Q12.1