### Mathematics Assignments | PDF | 8 to 12

PDF Files of Mathematics Assignments From VIII Standard to XII Standard PDF of mathematics Assignments for the students from VIII standard to XII standard.These assignments are strictly according to the CBSE and DAV Board Final question Papers

# Maths Assignment Class 12Vector 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  $\large \bg{black}\mathbf{\overrightarrow{a}-2\overrightarrow{b}+3\overrightarrow{c}}$,  $\large \bg{black}\mathbf{-2\overrightarrow{a}+3\overrightarrow{b}+2\overrightarrow{c}}$
and  $\large \bg{black}\mathbf{-8\overrightarrow{a}+13\overrightarrow{b}}$ 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 : $\large \bg{black}\mathbf{\overrightarrow{AB}=P.V.of B-P.V.of A}$

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  $\large \bg{black}\mathbf{\overrightarrow{a}=x\hat{i}+2\hat{j}+z\hat{k}}$  and
$\large \bg{black}\mathbf{\overrightarrow{b}=2\hat{i}+y\hat{j}+\hat{k}}$  are equal.

Ans: x = y = 2, x = 1

Question 5 :

If $\large \bg{black}\mathbf{\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k},\:\:\overrightarrow{b}=4\hat{i}-2\hat{j}+3\hat{k},\;\;\overrightarrow{c}=\hat{i}-2\hat{j}+\hat{k},}$ then find a vector of magnitude 6 units which is parallel to the vector  $\large \bg{black}\mathbf{ 2\overrightarrow{a}-\overrightarrow{b}+3\overrightarrow{c}.}$

Answer :  $\large \bg{black}\mathbf{\pm\: 2(\hat{i}-2\hat{j}+2\hat{k})}$
Question 6 :
Find a vector a of magnitude 5√2  square unit making an angle of  Ï€/4 with x - axis, Ï€/2 with the y - axis and an acute angle Î¸ with the z - axis
Answer:  $\large \bg{black}\mathbf{5\hat{i}+5\hat{k}}$

## Questions based of the scalar or dot product

Question 7 : If  |$\vec{a}$| = 3 and |$\vec{b}$| = 2 and $\bg{black}\vec{a}\:.\:\vec{b}{\color{DarkBlue}}$ = 4 then find the value of $\large \bg{black}\mathbf{(3\overrightarrow{a}-4\overrightarrow{b}).\;(2\overrightarrow{a}+5\overrightarrow{b})}$
Question 8: If the angle between two vectors $\large \bg{black}\mathbf{\overrightarrow{a}}$ and  $\large \bg{black}\mathbf{\overrightarrow{b}}$  of equal magnitude is  Ï€/6 and their dot product is $\large \bg{black}\mathbf{2\sqrt{3}}$ then find their magnitudes
Answer:   $\large \bg{black}\mathbf{|\overrightarrow{a}|=2=|\overrightarrow{b}|}$
Question 9: Find the cosine of the acute angle which the vector $\large \bg{black}\mathbf{\sqrt{2}\;\hat{i}+\hat{j}+\hat{k}}$ 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 Î» so that the projection of  $\large \bg{black}\mathbf{\overrightarrow{a}=\lambda\hat{i}+\hat{j}+4\hat{k}}$  on  $\large \bg{black}\mathbf{\overrightarrow{b}=2\hat{i}+6\hat{j}+3\hat{k}}$ is 4 units
Question 11:: Find $\large \bg{black}\bg{black}|\vec{a}-\vec{b}|$ if  $\large \bg{black}\bg{black}|\vec{a}|=2,\;|\vec{b}|=3$  and    $\large \bg{black}\bg{black}\vec{a}.\vec{b}$ = 4
Question 12 : If $\large \bg{black}\overrightarrow{a}$ and  $\large \bg{black}\overrightarrow{b}$  are two unit vectors and $\large \bg{black}\bg{black}|\vec{a}+\vec{b}|$ is also a unit vector then find the angle between $\large \bg{black}\overrightarrow{a}$ and  $\large \bg{black}\overrightarrow{b}$.

Question 13 : Let  $\large \bg{black}\overrightarrow{a},\;\overrightarrow{b},\;\overrightarrow{c}$  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:
$\large \bg{black}|\vec{a}+\vec{b}+\vec{c}|=5\sqrt{2}$
Question 14: If $\vec{a}+\vec{b}+\vec{c}=0$ and  $\bg{black}|\vec{a}|=5,\;|\vec{b}|=3$ and  $\bg{black}|\vec{c}|=7$ , then find the angle between
$\vec{a}$  and  $\vec{b}$
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 Î» by using  $\vec{d}.\vec{b}=0$
Question 16 Express the vector $\large \bg{black}\mathbf{\overrightarrow{a}=6\hat{i}-3\hat{j}-6\hat{k}}$  as the sum of two vectors such that one is parallel to the vector  $\large \bg{black}\mathbf{\overrightarrow{b}=\hat{i}+\hat{i}+\hat{k}}$  and the other is perpendicular to  $\vec{b}$
Answer: $\large \bg{black}\mathbf{-1(\hat{i}+\hat{j}+\hat{k})+7\hat{i}-2\hat{j}-5\hat{k}}$
Question 17: Dot product of a certain vector with vectors  $\large \bg{black}\large \bg{black}\mathbf{ 3\hat{i}-5\hat{k}$,  $\large \bg{black}\mathbf{2\hat{i}+7\hat{j}}$ and $\large \bg{black}\mathbf{\hat{i}+\hat{j}+\hat{k}}$  are respectively  -1, 6 and 5. Find the vector.
Answer:  $\large \bg{black}\mathbf{3\hat{i}+2\hat{k}}$
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  $\large \bg{black}\mathbf{\overrightarrow{a}=\hat{i}+2\hat{j}+3\hat{k}}$,  $\large \bg{black}\mathbf{\overrightarrow{b}=-\hat{i}+2\hat{j}+\hat{k}}$  and  $\large \bg{black}\mathbf{\overrightarrow{c}=3\hat{i}+\hat{j}}$  find Î» such that $\large \bg{black}\mathbf{\overrightarrow{a}+\lambda\overrightarrow{b}}$ is perpendicular to $\large \bg{black}\mathbf{\overrightarrow{c}}$

Hint : Two vectors are perpendicular if their dot product is zero
Question 19: Find the angle between $\large \bg{black}\mathbf{\overrightarrow{a}}$ and  $\large \bg{black}\mathbf{\overrightarrow{b}}$  if    $\large \bg{black}\mathbf{|\overrightarrow{a}|=\sqrt{3},}$  $\large \bg{black}\mathbf{|\overrightarrow{b}|=\sqrt{2}}$ and
$\large \bg{black}\mathbf{\overrightarrow{a}.\overrightarrow{b}=\sqrt{6}}$
Question 20
$\large \bg{black}\overrightarrow{a},\;\overrightarrow{b},\;\overrightarrow{c}$ are unit vectors, suppose
$\large \bg{black}\mathbf{\overrightarrow{a}\:.\:\overrightarrow{b}=\overrightarrow{a}\:.\:\overrightarrow{c}=0}$ and angle between
$\large \bg{black}\mathbf{\overrightarrow{b}}$ and $\large \bg{black}\mathbf{\overrightarrow{c}}$ is Ï€/6Then prove that $\large \bg{black}\mathbf{\overrightarrow{a}=\pm\: 2(\:\overrightarrow{b}\times\overrightarrow{c}\:)}$
Solution Hint:
$\large \bg{black}\mathbf{\overrightarrow{a}=\lambda\: (\:\overrightarrow{b}\times\overrightarrow{c}\:)}$
$\large \bg{black}\large \bg{black}\mathbf{|\overrightarrow{a}|=|\lambda|\: (\:|\overrightarrow{b}||\overrightarrow{c}|\:)sin\frac{\pi}{6}}$
⇒ Î» = åœŸ 2

## Questions based on the Vector or cross  product

Question 21
If  $\large \bg{black}\mathbf{\overrightarrow{p}=5\hat{i}+\lambda\hat{j}-3\hat{k}}$  and   $\large \bg{black}\mathbf{\overrightarrow{q}=\hat{i}+3\hat{j}-5\hat{k}}$ then find the value of Î» so that
$\large \bg{black}\mathbf{\overrightarrow{p}+\overrightarrow{q}}$ and  $\large \bg{black}\mathbf{\overrightarrow{p}-\overrightarrow{q}}$ are perpendicular vectors.
Question 22
Let $\large \bg{black}\mathbf{\overrightarrow{a}=\hat{i}+4\hat{j}+2\hat{k}}$,  $\large \bg{black}\mathbf{\overrightarrow{b}=3\hat{i}-2\hat{j}+7\hat{k}}$  and  $\large \bg{black}\mathbf{\overrightarrow{c}=2\hat{i}-\hat{j}+4\hat{k}}$ then find $\large \bg{black}\large \bg{black}\mathbf{\overrightarrow{d}$  which is perpendicular to both $\large \bg{black}\mathbf{\overrightarrow{a}}$  and  $\large \bg{black}\mathbf{\overrightarrow{b}}$  and $\large \bg{black}\mathbf{\overrightarrow{c}\:.\:\overrightarrow{d}=9}$
Answer:  $\large \bg{black}\mathbf{32\hat{i}-\hat{j}-14\hat{k}}$
Question 23
Let $\large \bg{black}\mathbf{\overrightarrow{a}=4\hat{i}+5\hat{j}-\hat{k}}$  , $\large \bg{black}\mathbf{\overrightarrow{b}=\hat{i}-4\hat{j}+5\hat{k}}$ and  $\large \bg{black}\mathbf{\overrightarrow{c}=3\hat{i}+\hat{j}-\hat{k}}$  then find  $\large \bg{black}\large \bg{black}\mathbf{\overrightarrow{d}$  which is perpendicular to both  $\large \bg{black}\large \bg{black}\mathbf{\overrightarrow{c}$  and $\large \bg{black}\mathbf{\overrightarrow{b}}$ and $\large \bg{black}\mathbf{\overrightarrow{d}\:.\:\overrightarrow{a}=2}$

Answer:   $\large \bg{black}\mathbf{\frac{-2}{63}\left(\hat{i}-16\hat{j}-13\hat{k}\right)}$
Question 24
Find a vector of magnitude 8 which is perpendicular to both vectors  $\large \bg{black}\mathbf{2\hat{i}-\hat{j}+3\hat{k}}$  and $\large \bg{black}\mathbf{\hat{i}+2\hat{j}-\hat{k}}$.
Answer:  $\large \bg{black}\mathbf{\frac{8}{\sqrt{35}}\left(-5\hat{i}-\hat{j}+3\hat{k}\right)}$

Question 25
If  $\large \bg{black}\mathbf{\overrightarrow{a}=3\hat{i}+2\hat{j}+2\hat{k}}$ and $\large \bg{black}\mathbf{\overrightarrow{b}=\hat{i}+2\hat{j}-2\hat{k}}$ then find a unit vector which is perpendicular to both the vectors $\large \bg{black}\mathbf{\overrightarrow{a}-\overrightarrow{b}}$  and  $\large \bg{black}\mathbf{\overrightarrow{a}+\overrightarrow{b}}$
Answer:   $\large \bg{black}\mathbf{\frac{1}{3}(-2\hat{i}+2\hat{j}+\hat{k})}$
Question 26
If  $\large \bg{black}\mathbf{\overrightarrow{a}=2\hat{i}+3\hat{j}-\hat{k}}$  , $\large \bg{black}\mathbf{\overrightarrow{b}=-4\hat{i}+3\hat{j}+2\hat{k}}$  and $\large \bg{black}\mathbf{\overrightarrow{c}=\hat{i}-2\hat{j}-3\hat{k}}$  then calculate  $\large \bg{black}\mathbf{(\overrightarrow{a}\times\overrightarrow{b}).(\overrightarrow{b}\times\overrightarrow{c})}$
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:   $\large \bg{black}\mathbf{\frac{1}{2}\sqrt{61}}$
Question 28
If  $\large \bg{black}\vec{a}\times\vec{b}=\vec{c}\times\vec{d}$ and $\large \bg{black}\vec{a}\times\vec{c}=\vec{b}\times\vec{d}$ then show that $\large \bg{black}\vec{a}-\vec{d}$  is parallel to $\large \bg{black}\vec{b}-\vec{c}$
Solution Hint:
Find $\large \bg{black}(\vec{a}-\vec{d})\times((\vec{b}-\vec{c})$  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 $\large \bg{black}\mathbf{10\sqrt{3}}$  units that are perpendicular to the plane of vectors  $\large \bg{black}\mathbf{\hat{i}+2\hat{j}+\hat{k}}$  and  $\large \bg{black}\mathbf{-\hat{i}+3\hat{j}+4\hat{k}}$
Answer: $\large \bg{black}\mathbf{10(\hat{i}-\hat{j}+\hat{k})}$

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.
Question 31
If  $|\vec{a}\times\vec{b}|^{2}+(\vec{a}.\vec{b})^{2}=400$  and  $|\vec{a}|=5$  , then find $|\vec{b}|$
Hint : Using Lagrange's identity here.

Question 32
If $\large \bg{black}\mathbf{\overrightarrow{a}=2\hat{i}-3\hat{j}+\hat{k}}$  , $\large \bg{black}\mathbf{\overrightarrow{b}=-\hat{i}+\hat{k}}$ and  $\large \bg{black}\mathbf{\overrightarrow{c}=2\hat{j}-\hat{k}}$ are three vectors, find the area of parallelogram having diagonals $\large \bg{black}\mathbf{\overrightarrow{a}+\overrightarrow{b}}$  and $\large \bg{black}\mathbf{\overrightarrow{b}+\overrightarrow{c}}$
Answer:  $\large \bg{black}\mathbf{{\sqrt{21}/2}$

Question 33
Two adjacent sides of a parallelogram are $\large \bg{black}\mathbf{\hat{i}-2\hat{j}-3\hat{k}}$  and  $\large \bg{black}\mathbf{2\hat{i}-4\hat{j}+5\hat{k}}$ . Find the unit vector parallel to one of its diagonals. Also find its area
Required unit vector is  $\large \bg{black}\mathbf{\frac{1}{3}(3\hat{i}-6\hat{j}+2\hat{k})}$  Area = $\large \bg{black}\mathbf{11\sqrt{5}}$  square unit.

## Questions based on the scalar triple  product

Note: These questions are deleted from CBSE syllabus
Question 34
Find Î» 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

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 $\vec{c}=2\hat{i}-\hat{j}+2\hat{k}$

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 Î»

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 $pqr-(p+q+r)$.

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 $|\vec{c}|^{2}$

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 Î» and then the required vector

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