Let two non-collinear unit vectors a ^ and b ^ form an acute angle. A point P moves, so that at any time t the position vector O P (where O is the origin) is given by a ^ cos t + b ^ sin t , when P is farthest from origin O , let M be the length of O P and u ^ be the unit vector along O P . Then,

u ^ = a ^ + b ^ a ^ + b ^ and M = 1 + a ^ · b ^ 1 2

u ^ = a ^ - b ^ a ^ - b ^ and M = 1 + a ^ · b ^ 1 2

u ^ = a ^ + b ^ a ^ + b ^ and M = 1 + 2 a ^ · b ^ 1 2

u ^ = a ^ - b ^ a ^ - b ^ and M = 1 + 2 a ^ · b ^ 1 2

Let the vectors P Q → , Q R → , R S → , S T → , T U → and U P → represent the sides of a regular hexagon. Statement I: P Q → × R S → + S T → ≠ 0 , because Statement II: P Q → × R S → = 0 and P Q → × S T → ≠ 0 .

Statement I is true, Statement II is false.

Statement I is true, Statement II is true and Statement II is a correct explanation for Statement I.

Statement I is true, Statement II is true but Statement II is not a correct explanation for Statement I.

Statement I is false, Statement II is true.

Let a → , b → , c → be unit vectors such that a → + b → + c → = 0 . Which one of the following is correct ?

a → × b → = b → × c → = c → × a → ≠ 0

a → × b → = b → × c → = c → × a → = 0

b → × b → = b → × c → = a → × c → = 0

a → × b → , b → × c → , c → × a → are mutually perpendicular

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The edges of a parallelepiped are of unit length and are parallel to non–coplanar unit vector $\hat{a}, \hat{b}, \hat{c}$ such that $\hat{a} . \hat{b} = \hat{b} . \hat{c} = \hat{a} . \hat{c} = \frac{1}{2}$ . Then, the volume of the parallelepiped is

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Important Questions on Product of Vectors

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JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 2 - Product of Vectors>Exercise for Session 8>Q 195

Let two non-collinear unit vectors

\hat{a}

and

\hat{b}

form an acute angle. A point

P

moves, so that at any time

t

the position vector

O P

(where

O

is the origin) is given by

\hat{a} \cos t + \hat{b} \sin t

, when

P

is farthest from origin

O

, let

M

be the length of

O P

and

\hat{u}

be the unit vector along

O P

. Then,

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JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 2 - Product of Vectors>Exercise for Session 8>Q 196

Let the vectors $\vec{P Q}, \vec{Q R}, \vec{R S}, \vec{S T}, \vec{T U}$ and $\vec{U P}$ represent the sides of a regular hexagon.

Statement I: $\vec{P Q} \times (\vec{R S} + \vec{S T}) \neq 0$ , because

Statement II: $\vec{P Q} \times \vec{R S} = 0$ and $\vec{P Q} \times \vec{S T} \neq 0$ .

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JEE Advanced

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Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 2 - Product of Vectors>Exercise for Session 8>Q 197

The number of distinct real values of

λ

, for which the vectors

- λ^{2} \hat{i} + \hat{j} + \hat{k}, \hat{i} - λ^{2} \hat{k} + \hat{j}

and

\hat{i} + \hat{j} - λ^{2} \hat{k}

are coplanar, is

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Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 2 - Product of Vectors>Exercise for Session 8>Q 198

Let

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

be unit vectors such that

\vec{a} + \vec{b} + \vec{c} = 0

. Which one of the following is correct

?

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JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 2 - Product of Vectors>Exercise for Session 8>Q 199

Let

\vec{A}

be vector parallel to the line of intersection of planes

P_{1}

and

P_{2}

through the origin.

P_{1}

is parallel to the vectors

2 \hat{j} + 3 \hat{k}

and

4 \hat{j} - 3 \hat{k}

and

P_{2}

is parallel to

\hat{j} - \hat{k}

and

3 \hat{i} + 3 \hat{j},

then the angle between the vectors

\vec{A}

and

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

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JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 2 - Product of Vectors>Exercise for Session 8>Q 200

Let,

\vec{a} = \overset{⏜}{i} + 2 \overset{⏜}{j} + \overset{⏜}{k}, \vec{b} = \overset{⏜}{i} - \overset{⏜}{j} + \overset{⏜}{k}, \vec{c} = \overset{⏜}{i} + \overset{⏜}{j} - \overset{⏜}{k}

. A vector coplanar to

\vec{a}

and

\vec{b}

has a projection along

\vec{c}

of magnitude

\frac{1}{\sqrt{3}}

, then the vector is

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JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 2 - Product of Vectors>Exercise for Session 8>Q 201

If $\vec{a}, \vec{b}, \vec{c}$ are three non-zero, non-coplanar vectors and

${\vec{b}}_{1} = \vec{b} - \frac{\vec{b} \cdot \vec{a}}{{|\vec{a}|}^{2}} \vec{a}, {\vec{b}}_{2} = \vec{b} + \frac{\vec{b} \cdot \vec{a}}{{|\vec{a}|}^{2}} \vec{a},$

${\vec{c}}_{1} = \vec{c} - \frac{\vec{c} \cdot \vec{a}}{{|\vec{a}|}^{2}} \vec{a} - \frac{\vec{c} \cdot \vec{b}}{{|\vec{b}|}^{2}} \vec{b}, {\vec{c}}_{2} = \vec{c} - \frac{\vec{c} \cdot \vec{a}}{{|\vec{a}|}^{2}} \vec{a} - \frac{\vec{c} \cdot {\vec{b}}_{1}}{{|\vec{b}|}^{2}} {\vec{b}}_{1},$

${\vec{c}}_{3} = \vec{c} - \frac{\vec{c} \cdot \vec{a}}{{|\vec{a}|}^{2}} \vec{a} - \frac{\vec{c} \cdot {\vec{b}}_{2}}{{|\vec{b}|}^{2}} {\vec{b}}_{2}, {\vec{c}}_{4} = \vec{a} - \frac{\vec{c} \cdot \vec{a}}{{|\vec{a}|}^{2}} \vec{a} .$

Then, which of the following is a set of mutually orthogonal vectors?

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JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 2 - Product of Vectors>Exercise for Session 8>Q 202

The unit vector which is orthogonal to the vector

3 \hat{i} + 2 \hat{j} + 6 \hat{k}

and is coplanar with the vectors

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

and

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

The edges of a parallelepiped are of unit length and are parallel to non–coplanar unit vector a^,b^,c^ such that a^.b^=b^.c^=a^.c^=12. Then, the volume of the parallelepiped is

Important Questions on Product of Vectors

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The edges of a parallelepiped are of unit length and are parallel to non–coplanar unit vector $\hat{a}, \hat{b}, \hat{c}$ such that $\hat{a} . \hat{b} = \hat{b} . \hat{c} = \hat{a} . \hat{c} = \frac{1}{2}$ . Then, the volume of the parallelepiped is