Mathematics Crash Course JEE Advanced>Chapter 26 - Vector Algebra>Exercise 1>Q 2

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The number of distinct real values of $λ$ for which the vectors $λ^{3} \hat{i} + \hat{k}$ , $\hat{i} - λ^{3} \hat{j}$ and $\hat{i} + (2 λ - \sin λ) \hat{j} - λ \hat{k}$ are coplanar, is/are

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Important Questions on Vector Algebra

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Mathematics Crash Course JEE Advanced>Chapter 26 - Vector Algebra>Exercise 1>Q 3

Points

X

and

Y

are taken on the sides

Q R

and

R S

, respectively of a parallelogram

P Q R S

, so that

\vec{Q X} = 4 \vec{X R}

and

\vec{R Y} = 4 \vec{Y S}

. If the line

X Y

cuts the line

P R

at

Z

, then

\vec{P Z}

is equal to

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Mathematics Crash Course JEE Advanced>Chapter 26 - Vector Algebra>Exercise 1>Q 4

If vector

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

is rotated through an angle of

90 °

, so as to cross the positive direction of

Y -

axis, then the vector in the new position is

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Mathematics Crash Course JEE Advanced>Chapter 26 - Vector Algebra>Exercise 1>Q 5

If

|x_{1}| > |y_{1}| + |z_{1}|, |y_{2}| > |x_{2}| + |z_{2}|, |z_{3}| > |x_{3}| + |y_{3}|

, then

x_{1} \hat{i} + y_{1} \hat{j} + z_{1} {\hat{k}}_{1}, x_{2} \hat{i} + y_{2} \hat{j} + z_{2} \hat{k}

and

x_{3} \hat{i} + y_{3} \hat{j} + z_{3} \hat{k}

are

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Mathematics Crash Course JEE Advanced>Chapter 26 - Vector Algebra>Exercise 1>Q 6

Let

A B C D

be a parallelogram such that

\vec{A B} = \vec{q}, \vec{A D} = \vec{p}

and

∠ B A D

be an acute angle. If

\vec{r}

is the vector that coincides with the altitude directed from the vertex

B

to the side

A D

, then

\vec{r}

is given by

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Mathematics Crash Course JEE Advanced>Chapter 26 - Vector Algebra>Exercise 1>Q 7

If

\vec{a}, \vec{b}

and

\vec{c}

are vectors such that

| \vec{a} | = 1, | \vec{b} | = 4, \vec{a} \cdot \vec{b} = 2

and

\vec{c} = (2 \vec{a} ⃗ \times \vec{b}) - 3 \vec{b}

, then the angle between

\vec{b}

and

\vec{c}

is

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Mathematics Crash Course JEE Advanced>Chapter 26 - Vector Algebra>Exercise 1>Q 8

If a vector

\vec{r}

is equally inclined with the vectors

\vec{a} = \cos θ \hat{i} + \sin θ \hat{j}, \vec{b} = - \sin θ \hat{i} + \cos θ \hat{j}

and

\vec{c} = \hat{k}

, then the angle between

\vec{r}

and

\vec{a}

is

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Mathematics Crash Course JEE Advanced>Chapter 26 - Vector Algebra>Exercise 1>Q 9

If in a

∆ A B C

,

\vec{B C} = \frac{\vec{e}}{|\vec{e}|} - \frac{\vec{f}}{|\vec{f}|}

and

\vec{A C} = \frac{2 \vec{e}}{|\vec{e}|}

;

|\vec{e}| \neq |\vec{f}|

, then the value of

\cos 2 A + \cos 2 B + \cos 2 C

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Mathematics Crash Course JEE Advanced>Chapter 26 - Vector Algebra>Exercise 1>Q 10

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

are three unit vectors,

\vec{a}

and

\vec{b}

are perpendicular to each other and vector

\vec{c}

is equally inclined to both

\vec{a}

and

\vec{b}

at an angle

θ

. If

\vec{c} = α \vec{a} + β \vec{b} + γ (\vec{a} \times \vec{b})

, where

α, β, γ

are constants, then