Let x → , y → and z → be three vectors each of magnitude 2 and the angle between each pair of them is π 3 . If a → is a non-zero vector perpendicular to x → and y → × z → and b → is a non-zero vector perpendicular to y → and z → × x → , then

a → b → = - ( a → · y → ) ( b → · z → )

a → = ( a → , y → ) ( z → - y → )

Let u ^ = u 1 i ^ + u 2 j ^ + u 3 k ^ be a unit vector in R 3 and w ^ = 1 6 i ^ + j ^ + 2 k ^ . Given that there exists a vector v → in R 3 such that u ^ × v → = 1 and w ^ . u ^ × v → = 1 . Which of the following statement(s) is (are) correct ?

If u ^ lies in the xy - plane then u 1 = u 2

There is exactly one choice for such v →

If u ^ lies in the xz - plane then 2 u 1 = u 3

Let a and b be positive real numbers. Suppose P Q → = a i ^ + b j ^ and P S → = a i ^ - b j ^ are adjacent sides of a parallelogram P Q R S . Let u → and v → be the projection vectors of w → = i ^ + j ^ along P Q → and P S → , respectively. If | u → | + v → = w → and if the area of the parallelogram P Q R S is 8 Sq. unit , then which of the following statements is/are TRUE?

The length of the diagonal P R → of the parallelogram P Q R S is 4 unit

w → is an angle bisector of the vectors P Q → and P S →

Let a and b be positive real numbers. Suppose P Q → = a i ^ + b j ^ and P S → = a i ^ - b j ^ are adjacent sides of a parallelogram P Q R S . Let u → and v → be the projection vectors of w → = i ^ + j ^ along P Q → and P S → , respectively. If | u → | + v → = w → and if the area of the parallelogram P Q R S is 8 Sq. unit , then which of the following statements is/are TRUE?

The length of the diagonal P R → of the parallelogram P Q R S is 4 unit

w → is an angle bisector of the vectors P Q → and P S →

Let u ^ = u 1 i ^ + u 2 j ^ + u 3 k ^ be a unit vector in R 3 and w ^ = 1 6 i ^ + j ^ + 2 k ^ . Given that there exists a vector v → in R 3 such that u ^ × v → = 1 and w ^ . u ^ × v → = 1 . Which of the following statement(s) is (are) correct ?

If u ^ lies in the xy - plane then u 1 = u 2

There is exactly one choice for such v →

If u ^ lies in the xz - plane then 2 u 1 = u 3

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The vector(s) which is/are coplanar with vectors $\hat{i} + \hat{j} + 2 \hat{k}$ and $\hat{i} + 2 \hat{j} + \hat{k},$ and perpendicular to the vector $\hat{i} + \hat{j} + \hat{k}$ is/are

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Important Questions on Vector Algebra

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Gamma Question Bank for Engineering Mathematics>Chapter 31 - Vector Algebra>Assignment>Q 14

Let

\vec{x}, \vec{y}

and

\vec{z}

be three vectors each of magnitude

\sqrt{2}

and the angle between each pair of them is

\frac{π}{3} .

\vec{a}

is a non-zero vector perpendicular to

\vec{x}

and

\vec{y} \times \vec{z}

and

\vec{b}

is a non-zero vector perpendicular to

\vec{y}

and

\vec{z} \times \vec{x},

then

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Gamma Question Bank for Engineering Mathematics>Chapter 31 - Vector Algebra>Assignment>Q 16

Let

\hat{u} = u_{1} \hat{i} + u_{2} \hat{j} + u_{3} \hat{k}

be a unit vector in

R^{3}

and

\hat{w} = \frac{1}{\sqrt{6}} (\hat{i} + \hat{j} + 2 \hat{k}) .

Given that there exists a vector

\vec{v}

R^{3}

such that

|\hat{u} \times \vec{v}| = 1

and

\hat{w} . (\hat{u} \times \vec{v}) = 1 .

Which of the following statement(s) is (are) correct ?

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Gamma Question Bank for Engineering Mathematics>Chapter 31 - Vector Algebra>Assignment>Q 17

Let

L_{1}

and

L_{2}

denote the lines

\vec{r} = \hat{i} + λ (- \hat{i} + 2 \hat{j} + 2 \hat{k}), λ \in ℝ

and

\vec{r} = μ (2 \hat{i} - \hat{j} + 2 \hat{k}), μ \in ℝ

respectively. If

L_{3}

is a line which is perpendicular to both

L_{1}

and

L_{2}

and cuts both of them, then which of the following options describe(s)

L_{3} ?

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Gamma Question Bank for Engineering Mathematics>Chapter 31 - Vector Algebra>Assignment>Q 18

Three lines

$L_{1} : \vec{r} = λ \hat{i} λ \in R$
$L_{2} : \vec{r} = \hat{k} + μ \hat{i}, μ \in R$
$L_{3} : \vec{r} = \hat{i} + \hat{j} + v \hat{k}, v \in R$

are given. For which point(s) $Q$ on $L_{2}$ can we find a point $P$ on $L_{1}$ and a point $R$ on $L_{3}$ so that $P, Q$ and $R$ are collinear?

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Gamma Question Bank for Engineering Mathematics>Chapter 31 - Vector Algebra>Assignment>Q 19

Let

a

and

b

be positive real numbers. Suppose

\vec{P Q} = a \hat{i} + b \hat{j}

and

\vec{P S} = a \hat{i} - b \hat{j}

are adjacent sides of a parallelogram

P Q R S .

Let

\vec{u}

and

\vec{v}

be the projection vectors of

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

along

\vec{P Q}

and

\vec{P S}

, respectively. If

| \vec{u} | + |\vec{v}| = |\vec{w}|

and if the area of the parallelogram

P Q R S

8 Sq. unit,

then which of the following statements is/are TRUE?

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Gamma Question Bank for Engineering Mathematics>Chapter 31 - Vector Algebra>Assignment>Q 24

Let

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

and

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

be three given vectors. If

r

is a vector such that

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

and

\vec{r} \cdot \vec{a} = 0

, then the value of

\vec{r} \cdot \vec{b}

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Gamma Question Bank for Engineering Mathematics>Chapter 31 - Vector Algebra>Assignment>Q 25

\vec{a}, \vec{b}

and

\vec{c}

are unit vectors satisfying

| \vec{a} - \vec{b} |^{2} + | \vec{b} - \vec{c} |^{2} + | \vec{c} - \vec{a} |^{2} = 9,

then

|2 \vec{a} + 5 \vec{b} + 5 \vec{c}|

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Gamma Question Bank for Engineering Mathematics>Chapter 31 - Vector Algebra>Assignment>Q 27

Let

\vec{a}, \vec{b}

and

\vec{c}

be three non-coplanar unit vectors such that the angle between every pair of them is

\frac{π}{3} .

\vec{a} \times \vec{b} + \vec{b} \times \vec{c} = p \vec{a} + q \vec{b} + r \vec{c},

where

p, q

and

r

are scalars, then the value of

\frac{p^{2} + 2 q^{2} + r^{2}}{q^{2}}

The vector(s) which is/are coplanar with vectors i^+j^+2k^ and i^+2j^+k^, and perpendicular to the vector i^+j^+k^ is/are

Important Questions on Vector Algebra

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The vector(s) which is/are coplanar with vectors $\hat{i} + \hat{j} + 2 \hat{k}$ and $\hat{i} + 2 \hat{j} + \hat{k},$ and perpendicular to the vector $\hat{i} + \hat{j} + \hat{k}$ is/are