Let A B C D be a parallelogram such that A B → = q → , A D → = p → and ∠ B A D be an acute angle. If r → is the vector that coincides with the altitude directed from the vertex B to the side A D , then r → is given by

r → = - q → + p → · q → p → · p → p →

r → = q → - p → · q → p → · p → p →

r → = 3 q → + 3 p → · q → p → · p → p →

r → = 3 q → - 3 p → · q → p → · p → p →

E M B I B E

Alpha Question Bank for Engineering: Mathematics>Chapter 20 - Vector Algebra>Exercise-3>Q 31

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If the vector $p \hat{i} + \hat{j} + \hat{k}, \hat{i} + q \hat{j} + \hat{k}$ and $\hat{i} + \hat{j} + r \hat{k} (p \neq q \neq r \neq 1)$ are coplanar, then the value of $p q r - (p + q + r)$ is-

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Important Questions on Vector Algebra

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Alpha Question Bank for Engineering: Mathematics>Chapter 20 - Vector Algebra>Exercise-3>Q 32

If

\vec{a}, \vec{b}

and

\vec{c}

are three non-zero vectors which are pairwise non-collinear. If

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

is collinear with

\vec{c}

and

\vec{b} + 2 \vec{c}

is collinear with

\vec{a}

then

\vec{a} + 3 \vec{b} + 6 \vec{c}

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 20 - Vector Algebra>Exercise-3>Q 33

Let

\hat{a}

and

\hat{b}

be two unit vectors. If the vectors

\vec{c} = \hat{a} + 2 \hat{b}

and

\vec{d} = 5 \hat{a} - 4 \hat{b}

are perpendicular to each other, then the angle between

\hat{a}

and

\hat{b}

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 20 - Vector Algebra>Exercise-3>Q 34

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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Alpha Question Bank for Engineering: Mathematics>Chapter 20 - Vector Algebra>Exercise-3>Q 35

The vectors

\vec{A B} = 3 \hat{i} + 4 \hat{k}

and

\vec{A C} = 5 \hat{i} - 2 \hat{j} + 4 \hat{k}

are sides of a triangle

A B C

. The length of the median through

A

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 20 - Vector Algebra>Exercise-3>Q 36

If

[\vec{a} \times \vec{b} \vec{b} \times \vec{c} \vec{c} \times \vec{a}]] = λ {[\vec{a} \vec{b} \vec{c}]}^{2}

, then $λ$ is equal to

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Alpha Question Bank for Engineering: Mathematics>Chapter 20 - Vector Algebra>Exercise-3>Q 37

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non-zero vectors such that no two of them are collinear and $(\vec{a} \times \vec{b}) \times \vec{c} = \frac{1}{3} |\vec{b}| |\vec{c}| \vec{a}$ . If $θ$ is the angle between vectors $\vec{b}$ and $\vec{c}$ , then the value of $\sin θ$ is

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Alpha Question Bank for Engineering: Mathematics>Chapter 20 - Vector Algebra>Exercise-3>Q 38

Let

\vec{a}, \vec{b}

and

\vec{c}

be three unit vectors such that

\vec{a} \times (\vec{b} \times \vec{c}) = \frac{\sqrt{3}}{2} (\vec{b} + \vec{c}) .

If

\vec{b}

is not parallel to

\vec{c}

, then the angle between

\vec{a}

and

\vec{b}

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 20 - Vector Algebra>Exercise-3>Q 39

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: