a → , b → , c → are three unit vectors, a → and b → are perpendicular to each other and vector c → is equally inclined to both a → and b → at an angle θ . If c → = α a → + β b → + γ a → × b → , where α , β , γ are constants, then

α = β = c o s θ , γ 2 = – c o s 2 θ

α = β = – cos θ , γ 2 = cos 2 θ

α = β = cos θ , r 2 = cos 2 θ

α = β = – c o s θ , 𝛾 2 = - c o s 2 θ

a → , b → , c → are three unit vectors, a → and b → are perpendicular to each other and vector c → is equally inclined to both a → and b → at an angle θ . If c → = α a → + β b → + γ a → × b → , where α , β , γ are constants, then

α = β = c o s θ , γ 2 = – c o s 2 θ

Let the unit vectors a → and b → be perpendicular to each other and the unit vector c → be inclined at an angle θ to both a → and b → . If c → = x a → + y b → + z a → × b → , then

x = y = cos θ , z 2 = - cos 2 θ

x = cos θ , y = sin θ , z = cos 2 θ

x = sin θ , y = cos θ , z = - cos 2 θ

x = y = cos θ , z 2 = cos 2 θ

If r → . a → = 0 , r → . b → = 1 and r → a → b → = 1 , a → . b → ≠ 0 , a → . b → 2 - a → 2 b → 2 = - 1 . Find value of r → in terms of a → and b → .

- a → × a → × b → + a → × b → a → × b → 2

a → × a → × b → + a → × b →

a → × b → × a → + b → × a → b → × a → 2

a → × a → × b → a → × a → × b → + a → × b →

If a ^ is a unit vector and projection of x → along a ^ is 2 units and a ^ × x → + b → = x → , then x → is equal to

1 2 2 a ^ + b → + a ^ × b →

1 2 a ^ – b → + a ^ × b →

If a → , b → and c → are any three non-zero vectors, then the component of a → × b → × c → perpendicular to b → is

a → × b → × c → + a → × b → · b → × c → b → 2 b →

a → × b → × c → + a → × b → · c → × a → b → 2 b →

a → × b → × c → + a → × c → · a → × b → b → 2 b →

a → × b → × c → + b → × c · b → × a → b → 2 b →

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

E M B I B E

Mathematics Crash Course JEE Advanced>Chapter 26 - Vector Algebra>Exercise 1>Q 10

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$\vec{a}, \vec{b}, \vec{c}$ are three unit vectors, $\vec{a}$ and $\vec{b}$ are perpendicular to each other and vector $\vec{c}$ is equally inclined to both $\vec{a}$ and $\vec{b}$ at an angle $θ$ . If $\vec{c} = α \vec{a} + β \vec{b} + γ (\vec{a} \times \vec{b})$ , where $α, β, γ$ are constants, then

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Important Questions on Vector Algebra

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Mathematics Crash Course JEE Advanced>Chapter 26 - Vector Algebra>Exercise 1>Q 11

Let the unit vectors

\vec{a}

and

\vec{b}

be perpendicular to each other and the unit vector

\vec{c}

be inclined at an angle

θ

to both

\vec{a}

and

\vec{b}

. If

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

, then

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Mathematics Crash Course JEE Advanced>Chapter 26 - Vector Algebra>Exercise 1>Q 12

In a tetrahedron

O A B C

, the edges are of lengths

|O A| = |B C| = a, |O B| = |A C| = b, |O C| = |A B| = c

. Let

G_{1}

and

G_{2}

be the centroids of the triangle

A B C

and

A O C

, respectively, such that

O G_{1} ⊥ B G_{2}

, then the value of

\frac{a^{2} + c^{2}}{b^{2}}

is

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Mathematics Crash Course JEE Advanced>Chapter 26 - Vector Algebra>Exercise 1>Q 13

Let two non-collinear vectors $\vec{a}$ and $\vec{b}$ inclined at an angle $\frac{2 π}{3}$ be such that $|\vec{a}| = 3$ and $|\vec{b}| = 2$ . If a point $P$ moves so that at any time $t$ its position vector $\vec{O P}$ , (where $O$ is the origin) is given as $\vec{O P} = (t + \frac{1}{t}) \vec{a} + (t - \frac{1}{t}) \vec{b}$ , then least distance of $P$ from the origin is