Scalar (Dot) Product

Interactive Feed

Let $a$ and $b$ be two unit vectors. If the vectors $c = a + 2 b$ and $d = 5 a - 4 b$ are perpendicular to each other, then the angle between $a$ and $b$ is:

\frac{π}{6}

\frac{π}{2}

\frac{π}{3}

\frac{π}{4}

Let $A B C D$ be a parallelogram such that $A B = q$ , $A D = p$ and $∠ B A D$ be an acute angle. If $r$ is the vector that coincide with the altitude directed from the vertex $B$ to the side $A D$ , then $r$ is given by:

r = 3 q - \frac{3 ( p \cdot q )}{( p \cdot p )} p

r = - q + \frac{( p \cdot q )}{( p \cdot p )} p

r = q - \frac{( p \cdot q )}{( p \cdot p )} p

r = - 3 q - \frac{3 ( p \cdot q )}{( p \cdot p )} p

If the vectors $a = \hat{i} - \hat{j} + 2 \hat{k}$ , $b = 2 \hat{i} + 4 \hat{j} + \hat{k}$ and $c = λ \hat{i} + \hat{j} + μ \hat{k}$ are mutually orthogonal, then $(λ, μ) =$

(2, - 3)

(- 2, 3)

(3, - 2)

(- 3, 2)

The values of $a$ , for which points $A, B, C$ with position vectors $2 \hat{i} - \hat{j} + \hat{k}, \hat{i} - 3 \hat{j} - 5 \hat{k}$ and $a \hat{i} - 3 \hat{j} + \hat{k}$ respectively are the vertices of a right angled triangle with $C = π /2$ are

- 2

- 1

- 2

- 1

A particles is acted upon by constant forces $4 \hat{i} + \hat{j} - 3 \hat{k}$ and $3 \hat{i} + \hat{j} - \hat{k}$ which displace it from a point $\hat{i} + 2 \hat{j} + 3 \hat{k}$ to the point $5 \hat{i} + 4 \hat{j} + \hat{k}$ . The work done in standard units by the forces is given by

Let $u, v, w$ be such that $∣ u ∣ = 1, ∣ v ∣ = 2, ∣ w ∣ = 3$ . If the projection $v$ along $u$ is equal to that of $w$ along $u$ and $v, w$ are perpendicular to each other then $∣ u - v + w ∣$ equals

7

14

A particle acted on by constant forces $4 \hat{i} + \hat{j} - 3 \hat{k}$ and $3 \hat{i} + \hat{j} - \hat{k}$ is displaced from the point $\hat{i} + 2 \hat{j} - 3 \hat{k}$ to the point $5 \hat{i} + 4 \hat{j} + \hat{k}$ . The total work done by the forces is

$a, b, c$ are 3 vectors, such that $a + b + c = 0$ , $∣ a ∣ = 1, ∣ b ∣ = 2, ∣ c ∣ = 3$ , then $a \cdot b + b \cdot c + c \cdot a$ is equal to

If $a, b, c$ are vectors show that $a + b + c = 0$ and $∣ a ∣ = 7, ∣ b ∣ = 5, ∣ c ∣ = 3$ then angle between vector $b$ and $c$ is

6 0^{\circ}

3 0^{\circ}

4 5^{\circ}

9 0^{\circ}

If $∣ a ∣ = 5, ∣ b ∣ = 4, ∣ c ∣ = 3$ thus what will be the value of $∣ a \cdot b + b \cdot c + c \cdot a ∣$ , given that $a + b + c = 0$

JEE Advanced 2011

Let $a = \hat{i} + \hat{j} + \hat{k}$ , $b = \hat{i} - \hat{j} + \hat{k}$ and $c = \hat{i} - \hat{j} - \hat{k}$ be three vectors. A vector $v$ in the plane of $a$ and $b$ , whose projection on $c$ is $\frac{1}{3}$ , is given by

\hat{i} - 3 \hat{j} + 3 \hat{k}

- 3 \hat{i} - 3 \hat{j} - \hat{k}

3 \hat{i} - \hat{j} + 3 \hat{k}

\hat{i} + 3 \hat{j} - 3 \hat{k}

JEE Advanced 2011

The vector(s) which is/are coplanar with vectors $\hat{i} + \hat{j} + 2 \hat{k}$ and $\hat{i} + 2 \hat{j} + \hat{k}$ , and perpendicular to the vector $\hat{i} + \hat{j} + \hat{k}$ is/are

* Multiple Correct Options

\hat{j} - \hat{k}

- \hat{i} + \hat{j}

\hat{i} - \hat{j}

- \hat{j} + \hat{k}

JEE Advanced 2010

Let $P, Q, R$ and $S$ be the points on the plane with position vectors $2 \hat{i} - \hat{j}, 4 \hat{i}, 3 \hat{i} + 3 \hat{j}$ and $\hat{i} + 2 \hat{j}$ respectively. The quadrilateral $P QR S$ must be a

parallelogram, which is neither a rhombus nor a rectangle

rectangle, but not a square

rhombus, but not a square

JEE Advanced 2010

LEVELJEE Advanced

Two adjacent sides of a parallelogram $A B C D$ are given by $A B = 2 \hat{i} + 10 \hat{j} + 11 \hat{k}$ and $A D = \hat{i} + 2 \hat{j} + 2 \hat{k}$ . The side $A D$ is rotated by an acute angle $α$ in the plane of the parallelogram so that $A D$ becomes $A D^{'}$ . If $A D^{'}$ makes a right angle with the side $A B$ , then the cosine of the angle $α$ is given by

\frac{8}{9}

\frac{17}{9}

\frac{1}{9}

\frac{4 5}{9}

JEE Advanced 2006

Let $a = \hat{i} + 2 \hat{j} + \hat{k}$ , $b = \hat{i} - \hat{j} + \hat{k}$ and $c = \hat{i} + \hat{j} - \hat{k}$ . A vector in the plane of $a$ and $b$ whose projection on $c$ is $\frac{1}{3}$ , is

4 \hat{i} - \hat{j} + 4 \hat{k}

3 \hat{i} + \hat{j} - 3 \hat{k}

2 \hat{i} + \hat{j} - 2 \hat{k}

4 \hat{i} + \hat{j} - 4 \hat{k}

JEE Advanced 2005S

If $a, b, c$ are three non-zero, non-coplanar vectors and $b_{1} = b - \frac{b \cdot a}{∣ a ∣ ^{2}} a$ , $b_{2} = b + \frac{b \cdot a}{∣ a ∣ ^{2}} a$ , $c_{1} = c - \frac{c \cdot a}{∣ a ∣ ^{2}} a - \frac{c \cdot b _{1}}{∣ b _{1} ∣ ^{2}} b_{1}$ , $c_{2} = c - \frac{c \cdot a}{∣ a ∣ ^{2}} a - \frac{c \cdot b}{∣ b ∣ ^{2}} b$ , $c_{3} = c - \frac{c \cdot a}{∣ a ∣ ^{2}} a - \frac{c \cdot b _{2}}{∣ b _{2} ∣ ^{2}} b_{2}$ , $c_{4} = c - \frac{c \cdot a}{∣ a ∣ ^{2}} a$ , then the set of orthogonal vectors is

(a, b_{1}, c_{3})

(a, b_{1}, c_{2})

(a, b_{1}, c_{1})

(a, b_{2}, c_{2})

JEE Advanced 2004S

The unit vector which is orthogonal to the vector $3 \hat{i} + 2 \hat{j} + 6 \hat{k}$ and is coplanar with the vectors $2 \hat{i} + \hat{j} + \hat{k}$ and $\hat{i} - \hat{j} + \hat{k}$ is

\frac{2 i ^ - 6 j ^ + k ^}{41}

\frac{2 i ^ - 3 j ^}{13}

\frac{3 i ^ - k ^}{10}

\frac{4 i ^ + 3 j ^ - 3 k ^}{34}

JEE Advanced 2002S

If $a$ and $b$ are two unit vectors such that $a + 2 b$ and $5 a - 4 b$ are perpendicular to each other then the angle between $a$ and $b$ is

4 5^{\circ}

6 0^{\circ}

cos^{- 1} (\frac{1}{3})

cos^{- 1} (\frac{2}{7})

JEE Advanced 2001S

If $a, b$ and $c$ are unit vectors, then $∣ a - b ∣^{2} + ∣ b - c ∣^{2} + ∣ c - a ∣^{2}$ does NOT exceed

4

9

8

6

JEE Advanced 1995S

Let $u, v$ and $w$ be vectors such that $u + v + w = 0$ . If $∣ u ∣ = 3, ∣ v ∣ = 4$ and $∣ w ∣ = 5$ , then $u \cdot v + v \cdot w + w \cdot u$ is

47

- 25

0

25

JEE Advanced 1994

Let $p$ and $q$ be the position vectors of $P$ and $Q$ respectively, with respect to $O$ and $∣ p ∣ = p, ∣ q ∣ = q$ . The points $R$ and $S$ divide $P Q$ internally and externally in the ratio $2 : 3$ respectively. If $O R$ and $O S$ are perpendicular then

9 q^{2} = 4 p^{2}

4 p^{2} = 9 q^{2}

9 p = 4 q

4 p = 9 q

JEE Advanced 1994

The vector $\frac{1}{3} (2 \hat{i} - 2 \hat{j} + \hat{k})$ is

* Multiple Correct Options

\frac{π}{3}

with the vector

(2 \hat{i} - 4 \hat{j} + 3 \hat{k})

parallel to the vector

(- \hat{i} + \hat{j} - \frac{1}{2} \hat{k})

perpendicular to the vector

3 \hat{i} + 2 \hat{j} - 2 \hat{k}

JEE Advanced 1993

Let $a = 2 \hat{i} - \hat{j} + \hat{k}$ , $b = \hat{i} + 2 \hat{j} - \hat{k}$ and $c = \hat{i} + \hat{j} - 2 \hat{k}$ be three vectors. A vector in the plane of $b$ and $c$ , whose projection on $a$ is of magnitude $2/3$ , is:

* Multiple Correct Options

2 \hat{i} + 3 \hat{j} - 3 \hat{k}

2 \hat{i} + 3 \hat{j} + 3 \hat{k}

- 2 \hat{i} - \hat{j} + 5 \hat{k}

2 \hat{i} + \hat{j} + 5 \hat{k}

JEE Advanced 2012

If $a, b$ and $c$ are unit vectors satisfying $∣ a - b ∣^{2} + ∣ b - c ∣^{2} + ∣ c - a ∣^{2} = 9$ , then $∣2 a + 5 b + 5 c ∣$ is .........

JEE Advanced 2005

If the incident ray on a surface is along the unit vector $\overset{v}{^}$ , the reflected ray is along the unit vector $\overset{w}{^}$ and the normal is along unit vector $\overset{a}{^}$ outwards. Express $\overset{w}{^}$ in terms of $\overset{a}{^}$ and $\overset{v}{^}$ .