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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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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Important Questions on Vector Algebra

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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}}

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Gamma Question Bank for Engineering Mathematics>Chapter 31 - Vector Algebra>Assignment>Q 28

Let

\vec{a}

and

\vec{b}

be two unit vectors such that

\vec{a} \cdot \vec{b} = 0 .

For some

x, y \in R,

let

\vec{c} = x \vec{a} + y \vec{b} + (\vec{a} \times \vec{b}) .

| \vec{c} | = 2

and the vector

\vec{c}

is inclined at the same angle

α

to both

\vec{a}

and

\vec{b},

then the value of

8 \cos^{2} α

is _____.

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Gamma Question Bank for Engineering Mathematics>Chapter 31 - Vector Algebra>Assignment>Q 29

Consider the cube in the first octant with sides

O P, O Q

and

O R

of length

1,

along the

x

-axis,

y

-axis and

z

-axis, respectively, where

O (0, 0, 0)

is the origin. Let

S (\frac{1}{2}, \frac{1}{2}, \frac{1}{2})

be the centre of the cube and

T

be the vertex of the cube opposite to the origin

O

such that

S

lies on the diagonal OT. If

\vec{p} = \vec{S P}, \vec{q} = \vec{S Q}, \vec{r} = \vec{S R}

and

\vec{t} = \vec{S T},

then the value of

2 | (\vec{p} \times \vec{q}) \times (\vec{r} \times \vec{t}) |

is _____.

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Gamma Question Bank for Engineering Mathematics>Chapter 31 - Vector Algebra>Assignment>Q 30

Three lines are given by

$\vec{r} = λ \hat{i}, λ \in R$ ,

$\vec{r} = μ (\hat{i} + \hat{j}), μ \in R$ and

$\vec{r} = ν (\hat{i} + \hat{j} + \hat{k}), ν \in R .$

Let the lines cut the plane $x + y + z = 1$ at the points $A, B$ and $C$ respectively. If the area of the triangle $A B C$ is $Δ$ then the value of ${(6 Δ)}^{2}$ equals_________

Let L1 and L2 denote the lines r→=i^+λ-i^+2j^+2k^, λ∈ℝ and r→=μ2i^-j^+2k^, μ∈ℝ respectively. If L3 is a line which is perpendicular to both L1and L2 and cuts both of them, then which of the following options describe(s) L3?

Important Questions on Vector Algebra

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