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

HARD

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If the volume of parallelepiped determined by vectors $(2 \bar{a} \times \bar{b}), (\bar{b} \times \bar{3 c})$ and $5 (\bar{c} \times \bar{a})$ is equal to the volume of the parallelepiped determined by vectors $5 (\bar{a} + \bar{b}), 6 (\bar{b} + \bar{c})$ and $2 (\bar{c} + \bar{a}),$ then find the volume of parallelepiped determined by vectors $\bar{a}, \bar{b} a n d \bar{c}$ in cubic units.

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Important Questions on Vectors

MEDIUM

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

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

and

\vec{c} = \hat{i} + m \hat{j} + n \hat{k}

are three coplanar vectors and

|\vec{c}| = \sqrt{6},

then which one of the following is correct?

MEDIUM

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

Let

x_{0}

be the point of local maxima of

f (x) = \vec{a} \cdot (\vec{b} \times \vec{c}),

where

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

and

\vec{c} = 7 \hat{i} - 2 \hat{j} + x \hat{k} .

Then the value of

\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}

x = x_{0}

is:

HARD

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

If the volume of a parallelepiped whose coterminous edges are

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

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

and

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

35

cu . m,

then a value of

\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} - \vec{c} \cdot \vec{a}

EASY

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

If the vectors

2 \hat{i} - 3 \hat{j} + 4 \hat{k}, 2 \hat{i} + \hat{j} - \hat{k}

and

λ \hat{i} - \hat{j} + 2 \hat{k}

are coplanar, then the value of

λ

HARD

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

If the volume of parallelepiped formed by the vectors

\hat{i} + λ \hat{j} + \hat{k}, \hat{j} + λ \hat{k}

and

λ \hat{i} + \hat{k}

is minimum, then

λ

is equal to:

HARD

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

\vec{a}, \vec{b}

and

\vec{c}

be three non-coplanar vectors and

\vec{p}, \vec{q},

and

\vec{r}

be defined by

\vec{p} = \frac{\vec{b} \times \vec{c}}{\vec{a} \cdot (\vec{b} \times \vec{c})}, \vec{q} = \frac{\vec{c} \times \vec{a}}{\vec{b} \cdot (\vec{c} \times \vec{a})}, \vec{r} = \frac{\vec{a} \times \vec{b}}{\bar{c} \cdot (\vec{a} \times \vec{b})}

such that

HARD

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

If the volume of a parallelopiped, whose coterminous edges are given by the vectors

\vec{a} = \hat{i} + \hat{j} + n \hat{k}

\vec{b} = 2 \hat{i} + 4 \hat{j} - n \hat{k}

and,

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

(n \geq 0)

158

cubic units, then :

MEDIUM

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

\vec{V} = 2 \hat{i} + \hat{j} - \hat{k}, \vec{W} = \hat{i} + 3 \hat{k}

and

\vec{U}

is a unit vector, then the maximum value of

[\begin{matrix} \vec{U} & \vec{V} & \vec{W} \end{matrix}]

HARD

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

A

B

C

and

D

are

(3, 7, 4), (5, - 2, - 3), (- 4, 5, 6)

and

(1, 2, 3)

respectively, then the volume of the parallelepiped with

A B

A C

and

A D

as the coterminous edges, is ….cubic units.

MEDIUM

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

Let the volume of a parallelepiped whose coterminous edges are given by

\vec{u} = \hat{i} + \hat{j} + λ \hat{k}, \vec{v} = \hat{i} + \hat{j} + 3 \hat{k}

and

\vec{w} = 2 \hat{i} + \hat{j} + \hat{k}

, be

1

cu. unit. If

θ

be the angle between the edges

\vec{u}

and

\vec{w},

then the value of

\cos θ

can be

MEDIUM

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

[\begin{array}{ll} \vec{a} & \vec{b} & \vec{c} \end{array}] = 4

, then the volume of the parallelopiped with coterminus edges

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

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

HARD

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

[\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

MEDIUM

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

The volume of a tetrahedron whose vertices are

4 \hat{i} + 5 \hat{j} + \hat{k}, - \hat{j} + \hat{k}, 3 \hat{i} + 9 \hat{j} + 4 \hat{k}

and

- 2 \hat{i} + 4 j + 4 \hat{k}

is (in cubic units)

MEDIUM

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

h

is the altitude of a parallelopiped determined by the vectors

\hat{a}, \hat{b}, \hat{c}

and the base is taken to be the parallelogram determined by

\hat{a}

and

\hat{b}

where

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

and

\hat{c} = \hat{i} + \hat{j} + 3 \hat{k},

then the value of

19 h^{2}

MEDIUM

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

\vec{a} = \frac{1}{\sqrt{10}} (3 \hat{ı} + \hat{k}), \vec{b} = \frac{1}{7} (2 \hat{ı} + 3 \hat{ȷ} - 6 \hat{k}),

then the value of

(2 \vec{a} - \vec{b}) \cdot [(\vec{a} \times \vec{b}) \times ((\vec{a} + 2 \vec{b})]

EASY

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

If the vectors

\vec{a} = \hat{ı} - 2 \hat{j} + \hat{k}

\vec{b} = 2 \hat{ı} - 5 \hat{j} + p \hat{k}

and

\vec{c} = 5 i - 9 \hat{j} + 4 \hat{k}

are coplanar, then the value of

p

EASY

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

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})

MEDIUM

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

[\vec{a} + 2 \vec{b} - \vec{c} \vec{a} - \vec{b} \vec{a} - \vec{b} - \vec{c}]

EASY

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

If the vectors,

\vec{p} = (a + 1) \hat{i} + a \hat{j} + a \hat{k}, \vec{q} = a \hat{i} + (a + 1) \hat{j} + a \hat{k}

and

\vec{r} = a \hat{i} + a \hat{j} + (a + 1) \hat{k} (a \in R)

are coplanar and

3 {(\vec{p} \cdot \vec{q})}^{2} - λ {|\vec{r} \times \vec{q}|}^{2} = 0

, then the value of

λ

is ________

HARD

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

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 ?

If the volume of parallelepiped determined by vectors (2a¯×b¯), (b¯×3c¯) and 5(c¯×a¯) is equal to the volume of the parallelepiped determined by vectors 5(a¯+b¯), 6(b¯+c¯) and 2(c¯+a¯), then find the volume of parallelepiped determined by vectors a¯, b¯ and c¯ in cubic units.

Important Questions on Vectors

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