Representation of Vectors

Q. a, b, c are three non coplanar , nonzero vectors , then prove that (a.a)bc +(a.b)ca +(a.c)ab =[ a b c ]a

Q. If a vector $\overrightarrow{AB}=2\hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{OB}=3\hat{i}-4\hat{j}+4\hat{k},$ then the position vector $\overrightarrow{OA}$ is

1. $3\hat{i}-2\hat{j}+\hat{k}$
2. $3\hat{i}-\hat{j}+\hat{k}$
3. $\hat{i}+3\hat{j}-2\hat{k}$
4. $\hat{i}-3\hat{j}+3\hat{k}$

Q.

Two or more vectors having the same initial point are called

1. zero vectors

2. unit vectors

3. co-initial vectors

4. co-terminus vectors.

Q.

Show that each of the given three vectors is a unit vector:

Also, show that they are mutually perpendicular to each other.

Q. Direction ratios of the line bisecting the acute angle between lines whose direction ratios are $(1,2,-1)$ and $(2,-1,-1),$ is

1. $(\frac{3}{\sqrt{6}},-\frac{1}{\sqrt{6}},-\frac{2}{\sqrt{6}})$
2. $(\frac{1}{\sqrt{6}},-\frac{3}{\sqrt{6}},-\frac{2}{\sqrt{6}})$
3. $(\frac{2}{\sqrt{6}},-\frac{1}{\sqrt{6}},-\frac{3}{\sqrt{6}})$
4. $(\frac{3}{\sqrt{6}},\frac{1}{\sqrt{6}},-\frac{2}{\sqrt{6}})$

Q. 14. Given =3 - j b=2+j-3k Express b = b1 +b2 where b1 is parallel to and b2 isperpendicular to (a, b, b1, b2 are vectors)

Q.

Let the vectors and be such that and, then is a unit vector, if the angle between and is

(A) (B) (C) (D)

Q. If the vectors $4\hat{i}-7\hat{j}-2\hat{k},\hat{i}+5\hat{j}-3\hat{k},3\hat{i}-\lambda \hat{j}+\hat{k},$ form a triangle, then the value of $\lambda$ is equal to

Q.

Find the angle between two vectorsandwith magnitudesand 2, respectively having.

Q.

A vector a has components $2\mathrm{p}$ and 1 w.r.t in a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter-clockwise sense. If w.r.t the new system, a has components $\mathrm{p}+1$ and $1$, then

Q. a and b are two non-collinear vectors, then xa+yb (where x and y are scalars) represents a vector which is

1. Parallel to b
2. Parallel to a
3. Coplanar with a and b
4. None of these

Q.

Find the direction cosines of the vector

Q. If $\overrightarrow{a}$and $\overrightarrow{b}$are two non-collinear vectors such that $x\overrightarrow{a}+y\overrightarrow{b}=\overrightarrow{0},$ then write the values of x and y.

Q.

ABCD is a parallelogram and $A_1$ and $B_1$ are the mid-points of the sides BC and CD respectively.
$If\overrightarrow{A{A}_1}+\overrightarrow{A{B}_1}=\lambda \overrightarrow{AC},$ then ${}^{\prime }{\lambda }^{\prime }$ is equal to

1. $\frac{1}{2}$

2. 1

3. $\frac{3}{2}$

4. 2

Q. Represent graphically a displacement of 40 km, 30° east of north.

Q. The square of magnitude of $3\hat{i}+4\hat{j}+5\hat{k}$ is

1. 25
2. 50
3. 15

Q.

If are mutually perpendicular vectors of equal magnitudes, show that the vector is equally inclined to and.

Q. If $a,b$ and $c$ are unit vectors then ${|a-b|}^2+{|b-c|}^2+{|c-a|}^2$ does not exceed

1. $4$
2. $9$
3. $8$
4. $6$

Q. The Square of the magnitude of the vector ​$3\hat{i}+4\hat{j}+5\hat{k}$ will be __

Q. If $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$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 $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$ are proportional
(d) both the vectors $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$have the same direction but different magnitudes

Q.

Find the angle between the vectors

Q. The square of the magnitude of the vector ​$3\hat{i}+4\hat{j}+5\hat{k}$ will be __

Q. Consider the vector $\overrightarrow{A}=2\hat{i}+3\hat{j}$. It's x component is units and y component is units.

1. 2
2. $\sqrt{13}$
3. $\frac{3}{2}$
4. 3

Q. a and b are two non-collinear vectors, then xa+yb (where x and y are scalars) represents a vector which is

1. None of these
2. Parallel to b
3. Coplanar with a and b
4. Parallel to a

Q.

Find the sum of the vectors.

Q.

ABCD is a parallelogram and $A_1$ and $B_1$ are the mid-points of the sides BC and CD respectively.
$If\overrightarrow{A{A}_1}+\overrightarrow{A{B}_1}=\lambda \overrightarrow{AC},$ then ${}^{\prime }{\lambda }^{\prime }$ is equal to

1. 1

2. 2

3. $\frac{1}{2}$

4. $\frac{3}{2}$

Q. In Figure ABCD is a regular hexagon, which vectors are:
(i) Collinear
(ii) Equal
(iii) Coinitial
(iv) Collinear but not equal.
Figure

Q. A vector $a$ has components $2p$ and $1$ with respect to a rectangular cartesian system. The system is rotated through a certain angle about the origin in the anti-clockwise sense. If $a$ has components $p+1$ and $1$ with respect to the new system, then

1. $p=0$
2. $p=1$ or $-\frac{1}{3}$
3. $p=-1$ or $\frac{1}{3}$
4. $p=1$ or $-1$

Q. A vector a has components 2p and 1 with respect to a rectangular cartesian system. The system is rotated through a certain angle about the origin in the anti-clockwise sense. If a has components p+1 and 1 with respect to the new system, then

1. $p=1$ or $-\frac{1}{3}$
2. $p=-1$ or $\frac{1}{3}$
3. $p=1$ or $-1$
4. $p=0$

Q.

Show that each of the given three vectors is a unit vector:

Also, show that they are mutually perpendicular to each other.