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Applications ...

Applications of Dot Product

Trending Questions

Q.

A force of magnitude of $30 N$ acting along $(\hat{i} + \hat{j} + \hat{k})$ , displaces a particle from a point $(2, 4, 1) t o (3, 5, 2)$ . The work done during this displacement is:

$90 J$
$30 J$
$30 \sqrt{3} J$
$20 J$

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Q.

If $a, b$ and $c > 0$ , then the minimum value of $\frac{a}{(b + c)} + \frac{b}{(c + a)} + \frac{c}{(a + b)}$ is

$1$
$\frac{3}{2}$
$2$
$\frac{5}{2}$

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

$\frac{π}{4}$
$\frac{π}{3}$
$\frac{π}{6}$
$\frac{π}{2}$

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Q. If, in a right angled triangle ABC, the hypotenuse AB = p, then AB . AC + BC . BA + CA . CB =

$2 p^{2}$
$\frac{p^{2}}{2}$
$p^{2}$
$N o n e$

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Q.

\sqrt[4]{\sqrt[3]{2^{2}}}

equals to

$2^{\frac{- 1}{6}}$
$2^{- 6}$
$2^{\frac{1}{6}}$
$2^{6}$

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Q. If

\to a

is perpendicular to

\to b

and

\to r

is a non – zero vector such that

p \to r + (\to r . \to b) \to a = \to c

, then

\to r

is equal to

$\frac{\to a}{p} - \frac{(\to c . \to a) \to b}{p^{2}}$
$\frac{\to c}{p} - \frac{(\to b . \to c) \to a}{p^{2}}$
$\frac{\to b}{p} - \frac{(\to a . \to c) \to c}{p^{2}}$
$\frac{\to c}{p^{2}} - \frac{(\to b . \to c) \to a}{p}$

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Q.

A spring of force constant $800 N m^{- 1}$ has an extension of $5 c m$ . The work done in extending it from $5 c m$ to $15 c m$ is

$16 J$
$8 J$
$32 J$
$24 J$

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Q. 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| =

2
14
$\sqrt{14}$
$\sqrt{7}$

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Q. If

x^{51} + 51

is divided by x+1, the remainder is

0
1
49
50

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Q. If |a|=3, |b|=4 and |a+b|=1, then |a-b|=

5
6
7
8

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Q.

If $4^{x}$ = 64, then

Positive
Negative
Zero
Fractional number

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Q. The unit vector in ZOX plane and making angle

45^{\circ}

and

60^{\circ}

respectively with

\to a = 2^i + 2^j -^k

and

\to b = 0^i +^j -^k

$- \frac{1}{\sqrt{2}}^i + \frac{1}{\sqrt{2}}^k$
$\frac{1}{\sqrt{2}}^i + \frac{1}{\sqrt{2}}^k$
$\frac{1}{3 \sqrt{2}}^i + \frac{4}{3 \sqrt{2}}^j + \frac{1}{3 \sqrt{2}}^k$
$N o n e o f t h e s e$

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Q. If

\to a, \to b, \to c

are vectors such that

\to a . \to b = 0

and

\to a + \to b = \to c

, then

$| \to a |^{2} + | \to b |^{2} = | \to c |^{2}$
$| \to a |^{2} = | \to b |^{2} + | \to c |^{2}$
$| \to b |^{2} + | \to a |^{2} = | \to c |^{2}$
$N o n e o f t h e s e$

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Q. The orthogonal projection of

3^i + 2^j - 5^k

on a vector perpendicular to

2^i -^j + 2^k

is

$\frac{13}{3}^i + \frac{4}{3}^j - \frac{11}{3}^k$
$\frac{13}{3}^i + \frac{4}{3}^j - \frac{11}{3}^k$
$\frac{13}{3}^i + \frac{4}{3}^j + \frac{11}{3}^k$
$\frac{13}{3}^i - \frac{4}{3}^j - \frac{11}{3}^k$

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Q. The length of the projection of

2^i - 3^j +^k

in the direction of

4^i - 4^j + 7^k

is

3
9
27
$3 \sqrt{3}$

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Q. Let

\to u, \to v

and

\to w

be vectors such that

\to u + \to v + \to w = \to 0 .

If

| \to u | = 3, | \to v | = 4

and

| \to w | = 5,

then

\to u . \to v + \to v . \to w + \to w . \to u

is

0
25
-25
47

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Q. If

a = -^i + 2^j + 3^k

and

b =^i - 2^j +^k

,

c = 4^i - 3^j - 2^k

and

a + λ b

is perpendicular to c, then

λ

=

1
3
4
2

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Q. If

| \to a | = | \to b | = | \to a + \to b | = 1, t h e n | \to a - \to b |

is equal to

1
$\sqrt{2}$
$\sqrt{3}$
None of these

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Q. If

a = 4^i + 6^j

and

b = 3^j + 4^k

, then the vector form of the component of a along b is

$\frac{18 (3^j + 4^k)}{25}$
$\frac{18 (3^j + 4^k)}{10 \sqrt{3}}$
$\frac{18 (3^j + 4^k)}{\sqrt{13}}$
$3^j +^k$

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Q. If a+b+c=0 and |a|=3, |b|=4 and

| c | = \sqrt{37}

, the angle between a and b is

$\frac{π}{4}$
$\frac{π}{2}$
$\frac{π}{6}$
$\frac{π}{3}$

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Q.

If $\to u = 3^i - 5^j + 9^k and \to v = 3^i + 4^j + 0 k;$ What is the component of $\to u$ along the direction of $\to v$ ?

$\frac{11}{7}$
$\frac{- 11}{5}$
$\frac{7}{11}$
$- 5$

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Q. A vector which makes equal angles with the vectors

\frac{1}{3} (^i - 2^j + 2^k), \frac{1}{5} (- 4^i - 3^k),^j .

is

$5^i +^j + 5^k$
$- 5^i +^j + 5^k$
$5^i -^j -^k$
$5^i +^j - 5^k$

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Q. If

\to a = 4^i + 6^j

and

\to b = 3 \to j + 4 \to k

then the vector form of component of

\to a

along

\to b

is

$\frac{18}{10 \sqrt{3}} (3^j + 4^k)$
$\frac{18}{25} (3^j + 4^k)$
$3^j + 4^k$
$\frac{18}{\sqrt{3}} (3^j + 4^k)$

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Q. If

| \to a | = 7, | \to b | = 11, | \to a + \to b | = 10 \sqrt{3}

, then

| \to a - \to b |

equals

10
$\sqrt{10}$
$2 \sqrt{10}$
20

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Q. Let

\to a = x^i + x^{2}^j + 2^k, \to b = - 3^i +^j +^k, \to c = (3 x + 11)^i + (x - 9)^j - 3^k

be three vectors such that the angle between

\to a

and

\to b

is acute and between

\to c

and

\to a

is obtuse, then x lies between

$(- \infty, 1) \cup (2, 3)$
$(- \infty, 0)$
(4, 5)
None of these

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Q. If

a^{x} = b^{y} = c^{z}

and

b^{2} = a c

then show that

y = \frac{2 z x}{z + x}

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Q.

cos $1^{\circ}$ $\times$ cos $2^{\circ}$ $\times$ cos $3^{\circ}$ $\times$ …….. $\times$ cos $180^{\circ}$ is equal to _________ .

(1/2)
1
0
-1

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Q. If

\to r . \to a = \to r . \to b = \to r . \to c = 0

where

\to a, \to b, \to c

are non – coplanar, then

$\to r ⊥ (\to b \times \to c)$
$\to r = \to 0$
$\to r ⊥ (\to c \times \to a)$
$\to r ⊥ (\to a \times \to a)$

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Q. The base of a triangle is divided into three equal parts. If

α, β, γ

be the angle subtended by these parts of vertex, prove that:

(cot α + cot β) (cot β + cot γ) = 4 c o s e c^{2} β

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Q. The work done by a force

\to F = 3^i - 4^j + 5^k

displaced the body from a point

(3, 4, 6)

to a point

(7, 2, 5)

is

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