Let α → = 3 i ^ + j ^ and β → = 2 i ^ - j ^ + 3 k ^ . If β → = β 1 → - β 2 → , where β 1 → is parallel to α → and β 2 → is perpendicular to α → , then β 1 → × β 2 → is equal to:

1 2 ( - 3 i ^ + 9 j ^ + 5 k ^ )

HARD

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Let $\vec{a} = 3 \hat{i} + 3 \hat{j} + 3 \hat{k}$ and $\vec{c} = \hat{j} - \hat{k}$ . Let $\vec{b}$ be such that $\vec{a} \cdot \vec{b} = 27$ and $\vec{a} \times \vec{b} = \vec{9 c}$

What is the angle between $(\vec{a} + \vec{b})$ and $\vec{c}$ ?

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Important Questions on Trigonometric Functions

MEDIUM

Mathematics>Trigonometry>Trigonometric Functions>Angles

Consider a triangular plot

A B C

with sides

A B = 7 m,

B C = 5 m

and

C A = 6 m .

A vertical lamp-post at the mid-point

D

A C

subtends an angle

30 °

B .

The height (in

m

) of the lamp-post is:

MEDIUM

Mathematics>Trigonometry>Trigonometric Functions>Angles

Let

\vec{a}, \vec{b}

and

\vec{c}

be three vectors such that

|\vec{a}| = \sqrt{3}, |\vec{b}| = 5, \vec{b} \cdot \vec{c} = 10

and the angle between

\vec{b}

and

\vec{c}

\frac{π}{3} .

\vec{a}

is perpendicular to the vector

\vec{b} \times \vec{c},

then

|\vec{a} \times (\vec{b} \times \vec{c})|

is equal to ____________.

HARD

Mathematics>Trigonometry>Trigonometric Functions>Angles

Let

\vec{α} = 3 \hat{i} + \hat{j}

and

\vec{β} = 2 \hat{i} - \hat{j} + 3 \hat{k} .

\vec{β} = \vec{β_{1}} - \vec{β_{2}},

where

\vec{β_{1}}

is parallel to

\vec{α}

and

\vec{β_{2}}

is perpendicular to

\vec{α},

then

\vec{β_{1}} \times \vec{β_{2}}

is equal to:

MEDIUM

Mathematics>Trigonometry>Trigonometric Functions>Angles

Let the position vectors of points

A

and

B

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

and

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

respectively. A point

P

divides the line segment

A B

internally in the ratio

λ : 1 (λ > 0)

. If

O

is the origin and

\vec{OB} \cdot \vec{OP} - 3 {|\vec{OA} \times \vec{OP}|}^{2} = 6

, then

λ

is equal to

HARD

Mathematics>Trigonometry>Trigonometric Functions>Angles

Given, $\vec{a} = 2 \hat{i} + \hat{j} - 2 \hat{k}$ and $\vec{b} = \hat{i} + \hat{j} .$ Let $\vec{c}$ be a vector such that $|\vec{c} - \vec{a}| = 3, |(\vec{a} \times \vec{b}) \times \vec{c}| = 3$ and the angle between $\vec{c}$ and $\vec{a} \times \vec{b}$ be $30 °$ . Then $\vec{a} \cdot \vec{c}$ is equal to:

EASY

Mathematics>Trigonometry>Trigonometric Functions>Angles

A unit vector perpendicular to the plane containing the vectors

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

and

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

HARD

Mathematics>Trigonometry>Trigonometric Functions>Angles

Let

\vec{a} = 3 \hat{i} + 2 \hat{j} + x \hat{k}

and

\vec{b} = \hat{i} - \hat{j} + \hat{k},

for some real

x

. Then

|\vec{a} \times \vec{b}| = r

is possible if

EASY

Mathematics>Trigonometry>Trigonometric Functions>Angles

A vector

\vec{a}

of length

2

units is making an angle

60 °

with each of the

X

-axis and

Y

-axis. If another vector

\vec{b}

of length

\sqrt{2}

units is making an angle

45 °

with each of the

Y

-axis and

Z

-axis, then

\vec{a} \times \vec{b} =

EASY

Mathematics>Trigonometry>Trigonometric Functions>Angles

|\vec{a}| = 13, |\vec{b}| = 5

and

\vec{a} \cdot \vec{b} = 30

,

then

|\vec{a} \times \vec{b}|

is equal to

HARD

Mathematics>Trigonometry>Trigonometric Functions>Angles

If the vector

\vec{b} = 3 \hat{j} + 4 \hat{k}

is written as the sum of a vector

\vec{b_{1}}

, parallel to

\vec{a} = \hat{i} + \hat{j}

and a vector

{\vec{b}}_{2},

perpendicular to

\vec{a},

then

{\vec{b}}_{1} \times {\vec{b}}_{2}

is equal to :

EASY

Mathematics>Trigonometry>Trigonometric Functions>Angles

{(\vec{a} \times \vec{b})}^{2} + {(\vec{a} \cdot \vec{b})}^{2} = 676

and

|\vec{b}| = 2

, then

|\vec{a}|

is equal to

MEDIUM

Mathematics>Trigonometry>Trigonometric Functions>Angles

A bird is sitting on the top of a vertical pole

20 m

high and its elevation from a point

O

on the ground is

45 °

. It flies off horizontally straight away from the point

O

. After one second, the elevation of the bird from

O

is reduced to

30 °

. Then the speed (in

m / s

) of the bird is

HARD

Mathematics>Trigonometry>Trigonometric Functions>Angles

Let

\hat{a}

and

\hat{b}

are two non-collinear unit vectors. If

\vec{u} = \hat{a} - (\hat{a} \cdot \hat{b}) \hat{b}

and

\vec{v} = \hat{a} \times \hat{b}

, then

| \vec{v} |

is equal to

EASY

Mathematics>Trigonometry>Trigonometric Functions>Angles

| \vec{a} | = 16, | \vec{b} | = 4

then,

\sqrt{| \vec{a} \times \vec{b} |^{2} + | \vec{a} \cdot \vec{b} |^{2}} =

HARD

Mathematics>Trigonometry>Trigonometric Functions>Angles

\vec{a} + l \vec{b} + l^{2} \vec{c} = 0

and

\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} = 3 (\vec{b} \times \vec{c}),

then the minimum value of such

l

MEDIUM

Mathematics>Trigonometry>Trigonometric Functions>Angles

\vec{a}, \vec{b}, \vec{c}

are non-zero non-collinear vectors and

\vec{a} \times \vec{b} = \vec{b} \times \vec{c} = \vec{c} \times \vec{a}

then

\vec{a} + \vec{b} + \vec{c} =

EASY

Mathematics>Trigonometry>Trigonometric Functions>Angles

\bar{O A} = \bar{i} + 2 \bar{j} + 3 \bar{k}

and

\bar{O B} = 4 \bar{i} + \bar{k}

are the position vectors of the points

A

and

B

, then the position vector of a point on the line passing through

B

and parallel to the vector

\bar{O A} \times \bar{O B}

which is at a distance of

\sqrt{189}

units from

B

HARD

Mathematics>Trigonometry>Trigonometric Functions>Angles

Let

Δ P Q R

be a triangle. Let

\vec{a} = \vec{Q R}, \vec{b} = \vec{R P} & \vec{c} = \vec{P Q}

. If

|\vec{a}| = 12, |\vec{b}| = 4 \sqrt{3}

and

\vec{b} \cdot \vec{c} = 24

, then which of the following is (are) true ?

HARD

Mathematics>Trigonometry>Trigonometric Functions>Angles

Consider the cube in the first octant with sides

O P, O Q

and

O R

of length

1

, along the

x

-axis,

y

-axis and

z

-axis, respectively, where

O (0, 0, 0)

is the origin. Let

S (\frac{1}{2}, \frac{1}{2}, \frac{1}{2})

be the centre of the cube and

T

be the vertex of the cube opposite to the origin

O

such that

S

lies on the diagonal

O T

. If

\vec{p} = \vec{S P}, \vec{q} = \vec{S Q}, \vec{r} = \vec{S R}

and

\vec{t} = \vec{S T}

, then the value of

|(\vec{p} \times \vec{q}) \times (\vec{r} \times \vec{t})|

is ______.

HARD

Mathematics>Trigonometry>Trigonometric Functions>Angles

If the angle of elevation of a cloud from a point

P

which is

25 m

above a lake be

30 °

and the angle of depression of reflection of the could in the lake from

P

60 °

, then the height of the cloud (in meters) from the surface of the lake is :

Let a→=3i^+3j^+3k^ and c→=j^-k^. Let b→ be such that a→·b→=27 and a→×b→=9c→ What is the angle between a→+b→ and c→?

Important Questions on Trigonometric Functions

Learn and Prepare for any exam you want !

Let $\vec{a} = 3 \hat{i} + 3 \hat{j} + 3 \hat{k}$ and $\vec{c} = \hat{j} - \hat{k}$ . Let $\vec{b}$ be such that $\vec{a} \cdot \vec{b} = 27$ and $\vec{a} \times \vec{b} = \vec{9 c}$

What is the angle between $(\vec{a} + \vec{b})$ and $\vec{c}$ ?