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, z = 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: θ = π/4 Required vector =   $\mathbf{5\hat{i}+5\hat{k}}$

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 parallel 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.

$\mathbf{|\overrightarrow{a}\times\overrightarrow{b}|^{2}+(\overrightarrow{a}.\overrightarrow{b})^{2}=|\overrightarrow{a}|^{2}|\overrightarrow{b}|^{2}$ 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 $\mathbf{\frac{1}{7}(3\hat{i}-6\hat{j}+2\hat{k})}$ Area = $\mathbf{11\sqrt{5}}$ square unit.

Question 34

A bird flies through a distance in a straight line given by the vector î+ 2ĵ+ k̂ . A man standing beside a straight metro rail track given by $\mathbf{\underset{r}{\rightarrow}$ = (3 + λ)î+ (2λ −1)ĵ+ 3λk̂ is observing the bird. Find the projected length of its flight on the metro track.

Ans: $\mathbf{8/\sqrt{14}}$

Question 35

The two vectors î + ĵ + k̂ and 3̂i − ĵ + 3k̂ represent the two sides OA and OB, respectively of a ∆OAB, where O is the origin. The point P lies on AB such that

OP is a median. Find the area of the parallelogram formed by the two adjacent

sides as OA and OP.

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

Solution Hint:

Algorithm

P is the mid point of AB, so find vector OP by using mid point formula.
Find the cross product between vector OP and vector OA.
Find the magnitude of vector OP and vector OA which is the area of parallelogram.

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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