The base vectors a → 1 , a → 2 and a → 3 are given in terms of base vectors b → 1 , b → 2 and b → 3 as a → 1 = 2 b → 1 + 3 b → 2 - b → 3 , a → 2 = b → 1 - 2 b → 2 + 2 b → 3 and a → 3 = - 2 b → 1 + b → 2 - 2 b → 3 . If F → = 3 b → 1 - b → 2 + 2 b → 3 , then vector F → in terms of a → 1 , a → 2 and a → 3 is

F → = 2 a → 1 + 5 a → 2 + 3 a → 3

F → = 3 a → 1 + 2 a → 2 + 5 a → 3

F → = 3 a → 1 - 5 a → 2 - 2 a → 3

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A unit tangent vector at $t = 2$ on the curve $x = t^{2} + 2$ , $y = 4 t - 5$ and $z = 2 t^{2} - 6 t$ is

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Important Questions on Introduction to Vectors

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IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Vectors and 3D Geometry JEE Main & Advanced>Chapter 1 - Introduction to Vectors>DPP>Q 20

If $\vec{a}$ and $\vec{b}$ are the position vectors of $A$ and $B$ , respectively, then the position vector of a point $C$ is $\vec{A B}$ , produced such that $\vec{A C} = 2015 \vec{A B}$ .

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Chapterwise/Topicwise Daily Practice Problems (DPP) Vectors and 3D Geometry JEE Main & Advanced>Chapter 1 - Introduction to Vectors>DPP>Q 21

Let

\vec{a} = (1, 1, - 1)

\vec{b} = (5, - 3, - 3)

and

\vec{c} = (3, - 1, 2)

. If

\vec{r}

is collinear with

\vec{c}

and has length of

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

, then

\vec{r}

equals

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JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Vectors and 3D Geometry JEE Main & Advanced>Chapter 1 - Introduction to Vectors>DPP>Q 22

A line passes through the points whose position vectors are $\hat{i} + \hat{j} - 2 \hat{k}$ and $\hat{i} - 3 \hat{j} + \hat{k}$ . The position vector of a point on the line at unit distance from the first point can be

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Chapterwise/Topicwise Daily Practice Problems (DPP) Vectors and 3D Geometry JEE Main & Advanced>Chapter 1 - Introduction to Vectors>DPP>Q 1

The number of integral values of

p

for which

(p + 1) \hat{i} - 3 \hat{j} + p \hat{k}, p \hat{i} + (p + 1) \hat{j} - 3 \hat{k}

and

- 3 \hat{i} + p \hat{j} + (p + 1) \hat{k}

are linearly dependent vectors is/are

MEDIUM

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Vectors and 3D Geometry JEE Main & Advanced>Chapter 1 - Introduction to Vectors>DPP>Q 2

The base vectors

{\vec{a}}_{1}, {\vec{a}}_{2}

and

{\vec{a}}_{3}

are given in terms of base vectors

{\vec{b}}_{1}, {\vec{b}}_{2}

and

{\vec{b}}_{3}

{\vec{a}}_{1} = 2 {\vec{b}}_{1} + 3 {\vec{b}}_{2} - {\vec{b}}_{3}, {\vec{a}}_{2} = {\vec{b}}_{1} - 2 {\vec{b}}_{2} + 2 {\vec{b}}_{3}

and

{\vec{a}}_{3} = - 2 {\vec{b}}_{1} + {\vec{b}}_{2} - 2 {\vec{b}}_{3}

. If

\vec{F} = 3 {\vec{b}}_{1} - {\vec{b}}_{2} + 2 {\vec{b}}_{3}

, then vector

\vec{F}

in terms of

{\vec{a}}_{1}, {\vec{a}}_{2}

and

{\vec{a}}_{3}

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JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Vectors and 3D Geometry JEE Main & Advanced>Chapter 1 - Introduction to Vectors>DPP>Q 3

The number of distinct real values of

λ

for which the vectors

λ^{3} \hat{i} + \hat{k}

\hat{i} - λ^{3} \hat{j}

and

\hat{i} + (2 λ - \sin λ) \hat{j} - λ \hat{k}

are coplanar, is/are

HARD

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Vectors and 3D Geometry JEE Main & Advanced>Chapter 1 - Introduction to Vectors>DPP>Q 4

The points

A (2 - x, 2, 2), B (2, 2 - y, 2), C (2, 2, 2 - z)

and

D (1, 1, 1)

are coplanar. Then the locus of

P (x, y, z)

HARD

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Vectors and 3D Geometry JEE Main & Advanced>Chapter 1 - Introduction to Vectors>DPP>Q 5

a_{1}

and

a_{2}

are two values of

a

for which the unit vector

a \hat{i} + b \hat{j} + \frac{1}{2} \hat{k}

is linearly dependent with

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

and

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

, then

\frac{1}{a_{1}} + \frac{1}{a_{2}}

is equal to

A unit tangent vector at t=2 on the curve x=t2+2, y=4t-5 and z=2t2−6t is

Important Questions on Introduction to Vectors

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A unit tangent vector at $t = 2$ on the curve $x = t^{2} + 2$ , $y = 4 t - 5$ and $z = 2 t^{2} - 6 t$ is