Assignment Class 12 Vector Algebra Chapter 10

Extra questions important for the examination. Maths assignment on vector algebra for complete knowledge of the concept.

Assignment on Vector Algebra
Question 1 :
Show that the points with position vector   $\mathbf{\overrightarrow{a}-2\overrightarrow{b}+3\overrightarrow{c}}$ ,   $\mathbf{-2\overrightarrow{a}+3\overrightarrow{b}+2\overrightarrow{c}}$
and   $\mathbf{-8\overrightarrow{a}+13\overrightarrow{b}}$ are collinear vectors.

Hint: Let given vectors are the position vectors of point A, B and C.
Find vector AB and AC. If one vector is the scalar multiple of the other then they are called parallel vectors.
Also these are co-initial vectors so these vector are called collinear vectors.
Question 2 : If A is a point (1, -2) and the vector AB has component 2 and 3, find the coordinates of point B.
Ans: (3, 1)
Hint : $\mathbf{\overrightarrow{AB}=P.V.of B-P.V.of A}$
Question 3 : Using vectors prove that the points A(-1, 2), B(0, 0), C(2, -4) are collinear.
Question 4 : Find the values of x, y, z so that the vector   $\mathbf{\overrightarrow{a}=x\hat{i}+2\hat{j}+z\hat{k}}$ and
$\mathbf{\overrightarrow{b}=2\hat{i}+y\hat{j}+\hat{k}}$ are equal.
Ans: x = y = 2, x = 1
Question 5 :
If $\mathbf{\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k},\:\:\overrightarrow{b}=4\hat{i}-2\hat{j}+3\hat{k},\;\;\overrightarrow{c}=\hat{i}-2\hat{j}+\hat{k},}$ then find a vector of magnitude 6 units which is parallel to the vector   $\mathbf{ 2\overrightarrow{a}-\overrightarrow{b}+3\overrightarrow{c}.}$
Answer :   $\mathbf{\pm\: 2(\hat{i}-2\hat{j}+2\hat{k})}$
Question 6 :
Find a vector a of magnitude 5√2 square unit making an angle of  π/4 with x - axis, π/2 with the y - axis and an acute angle θ with the z - axis
Answer:   $\mathbf{5\hat{i}+5\hat{k}}$

Questions based of the scalar or dot product

Question 7 : If | $\vec{a}$ | = 3 and | $\vec{b}$ | = 2 and $\vec{a}\:.\:\vec{b}{\color{DarkBlue}}$ = 4 then find the value of $\mathbf{(3\overrightarrow{a}-4\overrightarrow{b}).\;(2\overrightarrow{a}+5\overrightarrow{b})}$

Answer : 2

Question 8: If the angle between two vectors $\mathbf{\overrightarrow{a}}$ and $\mathbf{\overrightarrow{b}}$ of equal magnitude is π/6 and their dot product is $\mathbf{2\sqrt{3}}$ then find their magnitudes

Answer: $\mathbf{|\overrightarrow{a}|=2=|\overrightarrow{b}|}$

Question 9: Find the cosine of the acute angle which the vector $\mathbf{\sqrt{2}\;\hat{i}+\hat{j}+\hat{k}}$ makes with y-axis.

Ans: 1/2

Hint: Find the dot product between the given vector and the unit vector along y-axis.

Question 10: Find λ so that the projection of $\mathbf{\overrightarrow{a}=\lambda\hat{i}+\hat{j}+4\hat{k}}$ on $\mathbf{\overrightarrow{b}=2\hat{i}+6\hat{j}+3\hat{k}}$ is 4 units

Answer: 5

Question 11:: Find $|\vec{a}-\vec{b}|$ if $|\vec{a}|=2,\;|\vec{b}|=3$ and $\vec{a}.\vec{b}$ = 4

Answer: √5

Question 12 : If $\overrightarrow{a}$ and $\overrightarrow{b}$ are two unit vectors and $|\vec{a}+\vec{b}|$ is also a unit vector then find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ .

Answer: 2π/3

Question 13 : Let $\overrightarrow{a},\;\overrightarrow{b},\;\overrightarrow{c}$ be three vectors of magnitude 3, 4, 5 respectively. If each one is perpendicular to the sum of the other two vectors then prove that:

$|\vec{a}+\vec{b}+\vec{c}|=5\sqrt{2}$

Question 14: If $\vec{a}+\vec{b}+\vec{c}=0$ and $|\vec{a}|=5,\;|\vec{b}|=3$ and $|\vec{c}|=7$ , then find the angle between

$\vec{a}$ and $\vec{b}$

Answer : π/3

Hint: Bring vector c to the RHS and then squaring on both side

Question 15 : Express the vector $\vec{a}=5\hat{i}-2\hat{j}+5\hat{k}$ as the sum of two vectors such that one is parallel to the vector $\vec{b}=3\hat{i}+\hat{k}$ and the other is perpendicular to $\vec{b}$

Answer: $6\hat{i}+2\hat{k}+(-\hat{i}-2\hat{j}+3\hat{k})$

Hint : Let $\vec{a}=\vec{c}+\vec{d}$ find $\vec{c}=\lambda\;\vec{b}$ now find $\vec{d}$ by using $\vec{d}=\vec{a}-\vec{c}$ . Now find the value of λ by using $\vec{d}.\vec{b}=0$

Question 16 Express the vector $\mathbf{\overrightarrow{a}=6\hat{i}-3\hat{j}-6\hat{k}}$ as the sum of two vectors such that one is parallel to the vector $\mathbf{\overrightarrow{b}=\hat{i}+\hat{i}+\hat{k}}$ and the other is perpendicular to $\vec{b}$

Answer: $\mathbf{-1(\hat{i}+\hat{j}+\hat{k})+7\hat{i}-2\hat{j}-5\hat{k}}$

Question 17: Dot product of a certain vector with vectors $\mathbf{ 3\hat{i}-5\hat{k}$ , $\mathbf{2\hat{i}+7\hat{j}}$ and $\mathbf{\hat{i}+\hat{j}+\hat{k}}$ are respectively -1, 6 and 5. Find the vector.

Answer: $\mathbf{3\hat{i}+2\hat{k}}$

Hint : Consider a required vector in general form with components x, y, z, then find its dot product with all the given vectors and then solve the different equations.

Question 18: If $\mathbf{\overrightarrow{a}=\hat{i}+2\hat{j}+3\hat{k}}$ , $\mathbf{\overrightarrow{b}=-\hat{i}+2\hat{j}+\hat{k}}$ and $\mathbf{\overrightarrow{c}=3\hat{i}+\hat{j}}$ find λ such that $\mathbf{\overrightarrow{a}+\lambda\overrightarrow{b}}$ is perpendicular to $\mathbf{\overrightarrow{c}}$

Answer λ = 5

Hint : Two vectors are perpendicular if their dot product is zero

Question 19: Find the angle between $\mathbf{\overrightarrow{a}}$ and $\mathbf{\overrightarrow{b}}$ if $\mathbf{|\overrightarrow{a}|=\sqrt{3},}$ $\mathbf{|\overrightarrow{b}|=\sqrt{2}}$ and

$\mathbf{\overrightarrow{a}.\overrightarrow{b}=\sqrt{6}}$

Answer: 0

Question 20
$\overrightarrow{a},\;\overrightarrow{b},\;\overrightarrow{c}$ are unit vectors, suppose $\mathbf{\overrightarrow{a}\:.\:\overrightarrow{b}=\overrightarrow{a}\:.\:\overrightarrow{c}=0}$ and angle between

$\mathbf{\overrightarrow{b}}$ and $\mathbf{\overrightarrow{c}}$ is π/6. Then prove that $\mathbf{\overrightarrow{a}=\pm\: 2(\:\overrightarrow{b}\times\overrightarrow{c}\:)}$

Solution Hint:
$\mathbf{\overrightarrow{a}=\lambda\: (\:\overrightarrow{b}\times\overrightarrow{c}\:)}$

$\mathbf{|\overrightarrow{a}|=|\lambda|\: (\:|\overrightarrow{b}||\overrightarrow{c}|\:)sin\frac{\pi}{6}}$

⇒ λ = 土 2

Questions based on the Vector or cross product

Question 21

If $\mathbf{\overrightarrow{p}=5\hat{i}+\lambda\hat{j}-3\hat{k}}$ and $\mathbf{\overrightarrow{q}=\hat{i}+3\hat{j}-5\hat{k}}$ then find the value of λ so that

$\mathbf{\overrightarrow{p}+\overrightarrow{q}}$ and $\mathbf{\overrightarrow{p}-\overrightarrow{q}}$ are perpendicular vectors.

Answer: λ = 土 1

Question 22

Let $\mathbf{\overrightarrow{a}=\hat{i}+4\hat{j}+2\hat{k}}$ , $\mathbf{\overrightarrow{b}=3\hat{i}-2\hat{j}+7\hat{k}}$ and $\mathbf{\overrightarrow{c}=2\hat{i}-\hat{j}+4\hat{k}}$ then find $\mathbf{\overrightarrow{d}$ which is perpendicular to both $\mathbf{\overrightarrow{a}}$ and $\mathbf{\overrightarrow{b}}$ and $\mathbf{\overrightarrow{c}\:.\:\overrightarrow{d}=9}$

Answer: $\mathbf{32\hat{i}-\hat{j}-14\hat{k}}$

Question 23

Let $\mathbf{\overrightarrow{a}=4\hat{i}+5\hat{j}-\hat{k}}$ , $\mathbf{\overrightarrow{b}=\hat{i}-4\hat{j}+5\hat{k}}$ and $\mathbf{\overrightarrow{c}=3\hat{i}+\hat{j}-\hat{k}}$ then find $\mathbf{\overrightarrow{d}$ which is perpendicular to both $\mathbf{\overrightarrow{c}$ and $\mathbf{\overrightarrow{b}}$ and $\mathbf{\overrightarrow{d}\:.\:\overrightarrow{a}=2}$

Answer: $\mathbf{\frac{-2}{63}\left(\hat{i}-16\hat{j}-13\hat{k}\right)}$

Question 24

Find a vector of magnitude 8 which is perpendicular to both vectors $\mathbf{2\hat{i}-\hat{j}+3\hat{k}}$ and $\mathbf{\hat{i}+2\hat{j}-\hat{k}}$ .

Answer: $\mathbf{\frac{8}{\sqrt{35}}\left(-5\hat{i}-\hat{j}+3\hat{k}\right)}$

Question 25

If $\mathbf{\overrightarrow{a}=3\hat{i}+2\hat{j}+2\hat{k}}$ and $\mathbf{\overrightarrow{b}=\hat{i}+2\hat{j}-2\hat{k}}$ then find a unit vector which is perpendicular to both the vectors $\mathbf{\overrightarrow{a}-\overrightarrow{b}}$ and $\mathbf{\overrightarrow{a}+\overrightarrow{b}}$

Answer: $\mathbf{\frac{1}{3}(-2\hat{i}+2\hat{j}+\hat{k})}$

Question 26

If $\mathbf{\overrightarrow{a}=2\hat{i}+3\hat{j}-\hat{k}}$ , $\mathbf{\overrightarrow{b}=-4\hat{i}+3\hat{j}+2\hat{k}}$ and $\mathbf{\overrightarrow{c}=\hat{i}-2\hat{j}-3\hat{k}}$ then calculate $\mathbf{(\overrightarrow{a}\times\overrightarrow{b}).(\overrightarrow{b}\times\overrightarrow{c})}$

Answer: 45

Question 27

Using vectors find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5), C(1, 5, 5)

Answer: $\mathbf{\frac{1}{2}\sqrt{61}}$

Question 28

If $\vec{a}\times\vec{b}=\vec{c}\times\vec{d}$ and $\vec{a}\times\vec{c}=\vec{b}\times\vec{d}$ then show that $\vec{a}-\vec{d}$ is parallel to $\vec{b}-\vec{c}$

Solution Hint:

Find $(\vec{a}-\vec{d})\times((\vec{b}-\vec{c})$ then using the given conditions, if this product is = 0 then the vectors are perpendicular to each other.

Question 29

Find the vectors of magnitude $\mathbf{10\sqrt{3}}$ units that are perpendicular to the plane of vectors $\mathbf{\hat{i}+2\hat{j}+\hat{k}}$ and $\mathbf{-\hat{i}+3\hat{j}+4\hat{k}}$

Answer: $\mathbf{10(\hat{i}-\hat{j}+\hat{k})}$

Question 30

If $\vec{a}$ and $\vec{b}$ are unit vectors then find the angle between $\vec{a}$ and $\vec{b}$ if $\sqrt{3}\vec{a}-\vec{b}$ is a unit vector.

Answer: π/6

Question 31

If $|\vec{a}\times\vec{b}|^{2}+(\vec{a}.\vec{b})^{2}=400$ and $|\vec{a}|=5$ , then find $|\vec{b}|$

Answer: 4

Hint : Using Lagrange's identity here.

Question 32

If $\mathbf{\overrightarrow{a}=2\hat{i}-3\hat{j}+\hat{k}}$ , $\mathbf{\overrightarrow{b}=-\hat{i}+\hat{k}}$ and $\mathbf{\overrightarrow{c}=2\hat{j}-\hat{k}}$ are three vectors, find the area of parallelogram having diagonals $\mathbf{\overrightarrow{a}+\overrightarrow{b}}$ and $\mathbf{\overrightarrow{b}+\overrightarrow{c}}$

Answer: $\mathbf{{\sqrt{21}/2}$

Question 33

Two adjacent sides of a parallelogram are $\mathbf{\hat{i}-2\hat{j}-3\hat{k}}$ and $\mathbf{2\hat{i}-4\hat{j}+5\hat{k}}$ . Find the unit vector parallel to one of its diagonals. Also find its area

Answer:

Required unit vector is $\mathbf{\frac{1}{3}(3\hat{i}-6\hat{j}+2\hat{k})}$ Area = $\mathbf{11\sqrt{5}}$ square unit.

Questions based on the scalar triple product

Note: These questions are deleted from CBSE syllabus

Question 34

Find λ so that the four points with position vectors $3\hat{i}+2\hat{j}+\hat{k}$ , $4\hat{i}+\lambda\hat{j}+\;5\hat{k}$ , $4\hat{i}+2\hat{j}-2\hat{k}$ and $6\hat{i}+5\hat{j}-\hat{k}$ are coplanar

Answer : λ = 5

Question 35

Find the volume of the parallelopiped whose coterminous edges are $\vec{a}=2\hat{i}-3\hat{j}+4\hat{k}$ , $\vec{b}=\hat{i}+2\hat{j}-1\hat{k}$ and $\vec{c}=2\hat{i}-\hat{j}+2\hat{k}$

Answer: 2

Question 36

The volume of the parallelopiped whose coterminous edges are $\vec{a}=-12\hat{i}+\lambda\hat{k}$ , $\vec{b}=3\hat{j}-\hat{k}$ and $\vec{c}=2\hat{i}+\hat{j}-15\hat{k}$ is 546 cubic unit. Find λ

Answer : λ = -3

Question 37

(i) $[\lambda\vec{a}+\mu\vec{b}\;\;\vec{c}\;\;\vec{d}]=\lambda[\vec{a}\;\;\vec{b}\;\;\vec{c}]+\mu[\vec{b}\;\;\vec{c}\;\;\vec{d}]$

(ii) $[\vec{a}+\vec{b}\;\;\vec{b}+\vec{c}\;\;\vec{c}+\vec{a}]=2[\vec{a}\;\;\vec{b}\;\;\vec{c}]$

(iii) $[\vec{a}.(\vec{b}+\vec{c})\times(\vec{a}+2\vec{b}+3\vec{c})]=[\vec{a}\;\;\vec{b}\;\;\vec{c}]$

Question 38

If the vectors $p\hat{i}+\hat{j}+\hat{k}$ , $\hat{i}+q\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+r\hat{k}$ are coplanar then find the value of $pqr-(p+q+r)$ .

Answer: -2

Question 39

Let $\vec{a}=\hat{i}-\hat{j}$ , $\vec{b}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{c}$ is a vector such that $\vec{a}\times\vec{c}+\vec{b}=\vec{0}$ and $\vec{a}\:.\:\vec{c}=4$ then find the value of $|\vec{c}|^{2}$

Answer: 19/2

Solution Hint

Let $\vec{c}=x\hat{i}+y\hat{j}+z\hat{k}$ now using $\vec{a\times\vec{c}}=-\vec{b}$ and $\vec{a}\:.\:\vec{c}=4$ and by compairing the components find the relations between a, y, z. Solve these relations for the value of x, y, z, then find $\vec{c}$ and its magnitude.

Question 40

If $\vec{a}=\hat{i}+2\hat{j}+4\hat{k}$ , $\vec{b}=\hat{i}+\lambda\hat{j}+4\hat{k}$ and $\vec{c}=2\hat{i}+4\hat{j}+(\lambda^{2}-1)\hat{k}$ be coplanar vectors, then find the value of $\vec{a}\times\vec{c}$

Answer: $-10\hat{i}+5\hat{j}$

Question 41

Let $\vec{a}=3\hat{i}+2\hat{j}+2\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}-2\hat{k}$ be two vectors a vector perpendicular to both the vectors $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$ has the magnitude 12 then find the vector.

Answer: $\pm 4(2\hat{i}-2\hat{j}-\hat{k})$

Solution Hint: Required vector is $\lambda\left[(\vec{a}+\vec{b})\times(\vec{a}-\vec{b})\right]$ and $\left|\lambda\left[(\vec{a}+\vec{b})\times(\vec{a}-\vec{b})\right]\right|=12$ .

Find λ and then the required vector

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