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What are the value of the angles which a vector $\hat{i} + \hat{j} + \sqrt{2} \hat{k}$ makes with $X, Y$ and $Z$ axes respectively ?

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Important Questions on Vector Algebra

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Basics of Vectors

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

and

|\vec{b}| = 2

, then

|\vec{a}|

is equal to

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Basics of Vectors

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

then,

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

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Basics of Vectors

If the angle between

\vec{a}

and

\vec{b}

\frac{2 π}{3}

and the projection of

\vec{a}

in the direction of

\vec{b}

- 2,

then

| \vec{a} | =

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Basics of Vectors

\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

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Basics of Vectors

Let

\vec{a} = 2 \hat{i} + λ_{1} \hat{j} + 3 \hat{k}, \vec{b} = 4 \hat{i} + (3 - λ_{2}) \hat{j} + 6 \hat{k}

and

\vec{c} = 3 \hat{i} + 6 \hat{j} + (λ_{3} - 1) \hat{k}

be three vectors such that

\vec{b} = 2 \vec{a}

and

\vec{a}

is perpendicular to

\vec{c} .

Then a possible value of

(λ_{1}, λ_{2}, λ_{3})

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Basics of Vectors

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>Vectors and 3D Geometry>Vector Algebra>Basics of Vectors

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}

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Basics of Vectors

\hat{i}, \hat{j}, \hat{k}

are unit vectors and

{|\vec{a}|}^{2} = {\vec{a}}^{2}

, then the value of

{|\hat{i} \times \vec{a}|}^{2} + {|\hat{j} \times \vec{a}|}^{2} + {|\hat{k} \times \vec{a}|}^{2}

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Basics of Vectors

The magnitude of the projection of the vector

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

on the vector perpendicular to the plane containing the vectors

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

and

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

is:

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Basics of Vectors

\vec{P} and \vec{Q}

are two non-zero vectors inclined to each other at an angle

θ .

\hat{p} and \hat{q}

are unit vectors along

\vec{P} a n d \vec{Q}

respectively. The component of

\vec{Q}

in the direction of

\vec{P}

will be

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Basics of Vectors

The length of the projection of the line segment joining the points

(3, 4, 5)

and

(4, 6, 3)

on the line joining the points

(- 1, 2, 4)

and

(1, 0, 5)

MEDIUM

Mathematics>Vectors and 3D Geometry>Vector Algebra>Basics of Vectors

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

are vectors such that

\vec{a} + \vec{b} + \vec{c} = 0

and

|\vec{a}| = 7, |\vec{b}| = 5, |\vec{c}| = 3

then angle between vector

\vec{b}

and

\vec{c}

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Basics of Vectors

The unit vector in

Z O X

plane, making angles

45 °

and

60 °

respectively with

α = 2 \hat{i} + 2 \hat{j} - \hat{k}

and

β = \hat{j} - \hat{k}

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Basics of Vectors

θ

is the angle between the vectors

4 (\hat{i} - \hat{k})

and

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

then

(\sin θ + \cos θ)

equals to

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Basics of Vectors

Let

\vec{α}, \vec{β}, \vec{γ}

be the three unit vectors such that

\vec{α} \cdot \vec{β} = \vec{α} \cdot \vec{γ} = 0

and the angle between

\vec{β}

and

\vec{γ}

30 ° .

Then

\vec{α}

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Basics of Vectors

\vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{b} = \hat{i} - \hat{j} + \hat{k}, \vec{c} = \hat{i} + 2 \hat{j} - \hat{k}

then

|\begin{array}{l} \vec{a} \cdot \vec{a} & \bar{a} \cdot \vec{b} & \vec{a} \cdot \vec{c} \\ \vec{b} \cdot \vec{a} & \vec{b} \cdot \vec{b} & \vec{b} \cdot \bar{c} \\ \vec{c} \cdot \vec{a} & \vec{c} \cdot \vec{b} & \vec{c} \cdot \vec{c} \end{array}| =

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Basics of Vectors

\vec{a}

and

\vec{b}

are mutually perpendicular unit vectors, then

(3 \vec{a} + 2 \vec{b}) \cdot (5 \vec{a} - 6 \vec{b}) =

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Basics of Vectors

If $| \vec{a} | = 10, | \vec{b} | = 2$ and $\vec{a} \cdot \vec{b} = 12$ , then the value of $| \vec{a} \times \vec{b} |$ is

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Basics of Vectors

θ

is the angle between two vectors

\vec{a}

and

\vec{b},

then

\vec{a} \cdot \vec{b} \geq 0

only when

EASY

Mathematics>Vectors and 3D Geometry>Vector Algebra>Basics of Vectors

\vec{x} = 3 \hat{i} - 6 \hat{j} - \hat{k}, \vec{y} = \hat{i} + 4 \hat{j} - 3 \hat{k}

and

\vec{z} = 3 \hat{i} - 4 \hat{j} - 12 \hat{k}

, then the magnitude of the projection of

\vec{x} \times \vec{y}

\vec{z}

What are the value of the angles which a vector i^+j^+2k^ makes with X, Y and Z axes respectively ?

Important Questions on Vector Algebra

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What are the value of the angles which a vector $\hat{i} + \hat{j} + \sqrt{2} \hat{k}$ makes with $X, Y$ and $Z$ axes respectively ?