If A , B , C are three non-collinear points with position vectors a → , b → , c → , respectively, show that the length of the perpendicular from C on A B is a → × b → + b → × c → + c → × a → b → - a →

Consider △ A B C with side A B So, length of the perpendicular from C on A B is the height of the triangle, h We know that area of the triangle = 1 2 × base × height ⇒ h = Area of the triangle 1 2 A B ⇒ h = 1 2 A B → × A C → 1 2 A B → ⇒ h = b → - a → × c → - a → b → - a → ⇒ h = b → × c → - b → × a → - a → × c → + a → × a → b → - a → ⇒ h = a → × b → + b → × c → + c → × a → b → - a → Hence proved.

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If $|\vec{a} \times \vec{b}| = a b$ then the angle between $\vec{a} and \vec{b}$ is

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Important Questions on Vectors, Scalars and Elementary Calculus

EASY

Physics>Mechanics>Vectors, Scalars and Elementary Calculus>Scalars and Vectors

\vec{P} \times \vec{Q} = \vec{Q} \times \vec{P}

, the angle between

\vec{P}

and

\vec{Q}

θ (0 ° < θ < 360 °)

. The value of

θ

will be ___

°

HARD

Physics>Mechanics>Vectors, Scalars and Elementary Calculus>Scalars and Vectors

\vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{b} = 2 \hat{i} - \hat{j} + 3 \hat{k}

and

\vec{c} = \hat{i} - \hat{j}

and if

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

then

(λ_{1}, λ_{2}, λ_{3}) =

MEDIUM

Physics>Mechanics>Vectors, Scalars and Elementary Calculus>Scalars and Vectors

The direction ratios of the line which is perpendicular to lines

\frac{x - 1}{2} = \frac{y + 17}{- 3} = \frac{z - 6}{1}

and

\frac{x + 5}{1} = \frac{y + 3}{2} = \frac{z - 4}{- 2}

are

HARD

Physics>Mechanics>Vectors, Scalars and Elementary Calculus>Scalars and Vectors

Let

\vec{a} = 2 \hat{i} - 3 \hat{j} + 4 \hat{k}

and

\vec{b} = 7 \hat{i} + \hat{j} - 6 \hat{k}

\vec{r} \times \vec{a} = \vec{r} \times \vec{b}, \vec{r} \cdot (\hat{i} + 2 \hat{j} + \hat{k}) = - 3,

then

\vec{r} \cdot (2 \hat{i} - 3 \hat{j} + \hat{k})

is equal to:

MEDIUM

Physics>Mechanics>Vectors, Scalars and Elementary Calculus>Scalars and Vectors

The area of the parallelogram, whose diagonals are

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

and

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

is equal to

EASY

Physics>Mechanics>Vectors, Scalars and Elementary Calculus>Scalars and Vectors

If the angle between the vectors

\vec{P}

and

\vec{Q}

θ,

the value of

(\vec{Q} \times \vec{P}) \cdot \vec{P}

EASY

Physics>Mechanics>Vectors, Scalars and Elementary Calculus>Scalars and Vectors

Consider the vectors

\vec{A} = (\hat{i} + \hat{j} - \hat{k})

\vec{B} = (2 \hat{i} - \hat{j} + \hat{k})

and

\vec{C} = \frac{1}{\sqrt{5}} (\hat{i} - 2 \hat{j} + 2 \hat{k})

. What is the value of

\vec{C} \cdot (\vec{A} \times \vec{B})

EASY

Physics>Mechanics>Vectors, Scalars and Elementary Calculus>Scalars and Vectors

If the area of the parallelogram with

\vec{a}

and

\vec{b}

as two adjacent sides is

15

sq. units then the area of the parallelogram having

3 \vec{a} + 2 \vec{b}

and

\vec{a} + 3 \vec{b}

as two adjacent sides in sq. units is

MEDIUM

Physics>Mechanics>Vectors, Scalars and Elementary Calculus>Scalars and Vectors

If the vertices of

Δ ABC

are

A = (2, 3, 5), B = (- 1, 3, 2), C = (3, 5, - 2),

then the area of the

Δ ABC

(in sq. units) is

HARD

Physics>Mechanics>Vectors, Scalars and Elementary Calculus>Scalars and Vectors

Let

D

and

E

be the midpoints of the sides

A C

and

B C

of a triangle

A B C

respectively. If

O

is an interior point of the triangle

A B C

such that

\vec{O A} + 2 \vec{O B} + 3 \vec{O C} = \vec{0},

then the area (in sq. units) of the triangle

O D E

MEDIUM

Physics>Mechanics>Vectors, Scalars and Elementary Calculus>Scalars and Vectors

\vec{a}

is a unit vector, then

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

EASY

Physics>Mechanics>Vectors, Scalars and Elementary Calculus>Scalars and 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:

MEDIUM

Physics>Mechanics>Vectors, Scalars and Elementary Calculus>Scalars and Vectors

\vec{A}

and

\vec{B}

are two vectors satisfying the relation

\vec{A} \cdot \vec{B} = | \vec{A} \times \vec{B} |

. Then the value of

| \vec{A} - \vec{B} |

will be:

HARD

Physics>Mechanics>Vectors, Scalars and Elementary Calculus>Scalars and Vectors

Let

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

and

\vec{b} = 3 \hat{i} - α \hat{j} + \hat{k}

. If the area of the parallelogram whose adjacent sides are represented by the vectors

\vec{a}

and

\vec{b}

8 \sqrt{3}

square units, then

\vec{a} \cdot \vec{b}

is equal to ___ .

MEDIUM

Physics>Mechanics>Vectors, Scalars and Elementary Calculus>Scalars and Vectors

The area of the quadrilateral

A B C D,

where

A (0, 4, 1), B (2, 3, - 1), C (4, 5, 0)

and

D (2, 6, 2)

is equal to

MEDIUM

Physics>Mechanics>Vectors, Scalars and Elementary Calculus>Scalars and Vectors

A, B, C

are three non-collinear points with position vectors

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

, respectively, show that the length of the perpendicular from

C

A B

\frac{|(\vec{a} \times \vec{b}) + (\vec{b} \times \vec{c}) + (\vec{c} \times \vec{a})|}{|\vec{b} - \vec{a}|}

EASY

Physics>Mechanics>Vectors, Scalars and Elementary Calculus>Scalars and Vectors

\vec{a} = 2 \hat{i} + 2 \hat{j} + \hat{k}, \vec{|b|} = 6

and the angle between

\vec{a}

and

\vec{b}

\frac{π}{6},

then the area of the triangle (in square units) with

\bar{a}

and

\bar{b}

as two of its sides is

MEDIUM

Physics>Mechanics>Vectors, Scalars and Elementary Calculus>Scalars and Vectors

\vec{a} = 2 \hat{i} + \hat{j} - 3 \hat{k}, \vec{b} = \hat{i} - 2 \hat{j} + \hat{k}, \vec{c} = - \hat{i} + \hat{j} - 4 \hat{k}

and

\bar{d} = \hat{i} + \hat{j} + \hat{k},

then

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

HARD

Physics>Mechanics>Vectors, Scalars and Elementary Calculus>Scalars and Vectors

Let

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

and

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

be two vectors. If a vector perpendicular to both the vectors

\vec{a} + \vec{b}

and

\vec{a} - \vec{b}

has the magnitude

12

then one such vector is:

EASY

Physics>Mechanics>Vectors, Scalars and Elementary Calculus>Scalars and Vectors

Let

\vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{b} = \hat{i} + 3 \hat{j} + 5 \hat{k}

and

\vec{c} = 7 \hat{i} + 9 \hat{j} + 11 \hat{k}

.

Then, the area of the parallelogram with diagonals

\vec{a} + \vec{b}

and

\vec{b} + \vec{c}

If a→×b→=a b then the angle between a→ and b→ is

Important Questions on Vectors, Scalars and Elementary Calculus

Learn and Prepare for any exam you want !

If $|\vec{a} \times \vec{b}| = a b$ then the angle between $\vec{a} and \vec{b}$ is