Mathematics · JEE

Scalar and vector products Mock Test for JEE

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Syllabus context

Part of Vector Algebra in JEE Main Mathematics.

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Q1MathsUnit 12: Vector Algebra

Let

a, \beta, \gamma

be distinct real numbers. The points with position vectors ai

+\beta j+

\gamma k, \beta i+\gamma j+a k, \gamma i+a j+\beta k

Q2MathsUnit 12: Vector Algebra

The non zero vector

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

related by

\vec{a}=8 \vec{b}

and

\vec{c}=-7 \vec{b},

then angle between

\vec{a} \& \vec{c}

Q3MathsUnit 12: Vector Algebra

Consider

\Delta A B C

with

A \equiv(\vec{a}), B \equiv(\vec{b})

and

C=(\vec{c}),

\vec{b} .(\vec{a}+\vec{c})=\vec{b} \cdot \vec{b}+

\vec{a} \cdot \vec{c} ;|\vec{b}-\vec{a}|=3 ;|\vec{c}-\vec{b}|=4

then the angle between the medians

\overline{A M}

and

B D

Q4MathsUnit 12: Vector Algebra

If position vectors of four vertices of a square are

\boldsymbol{A}=\mathbf{3} \boldsymbol{i}-\boldsymbol{\lambda} \boldsymbol{j}, \boldsymbol{B}=\mathbf{9} \boldsymbol{i}-\boldsymbol{6} \boldsymbol{k}

\boldsymbol{C}=\mathbf{6} \boldsymbol{j}-\mathbf{5} \boldsymbol{k}

and

\boldsymbol{D}=\boldsymbol{\alpha} \boldsymbol{i}-\boldsymbol{3} \boldsymbol{j}+\boldsymbol{4} \boldsymbol{k}

then value of

\lambda-\alpha

is :

Q5MathsUnit 12: Vector Algebra

For any four points

P, Q, R, S

|\overrightarrow{P Q} \times \overrightarrow{\boldsymbol{R}} \boldsymbol{S}-\overrightarrow{\boldsymbol{Q} \boldsymbol{R}} \times \overrightarrow{\boldsymbol{P}} \boldsymbol{S}+\overrightarrow{\boldsymbol{R}} \boldsymbol{P} \times \overrightarrow{\boldsymbol{Q}} \boldsymbol{S}|

is equal to 4 times the area of the triangle.

Q6MathsUnit 12: Vector Algebra

\boldsymbol{A}=(\mathbf{1}, \mathbf{2}, \mathbf{3}), \boldsymbol{B}=(\mathbf{2}, \mathbf{3}, \mathbf{4})

and

\boldsymbol{A} \boldsymbol{B}

is produced upto

C

such that

2 A B=B C

then

C=

Q7MathsUnit 12: Vector Algebra

a, b, c

are non-coplanar vectors and

\lambda

is a real number, then

\left[\lambda(\vec{a}+\vec{b}) \lambda^{2} \vec{b} \lambda \vec{c}\right]=[\vec{a} \quad \vec{b}+\vec{c} \quad \vec{b}]

for

Q8MathsUnit 12: Vector Algebra

The points

A(-1,3,0), B(2,2,1)

and

C(1,1,3)

determine a plane. The distance of the plane

A, B, C

from the point

D(5,7,8)

Q9MathsUnit 12: Vector Algebra

The vectors

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

and

\mathbf{3} \hat{\mathbf{i}}+\hat{\boldsymbol{j}}+\mathbf{4} \hat{\boldsymbol{k}}

are so placed that the end point of one vector is the starting point of the next vector. Then the vectors are :

Q10MathsUnit 12: Vector Algebra

\vec{V}=3 \hat{i}+4 \hat{j}

then, with what scalar 'C must it be multiplied so that

C|\overrightarrow{\boldsymbol{V}}|=

\mathbf{7 . 5}:

Q11MathsUnit 12: Vector Algebra

The vectors

\overrightarrow{A B}=3 \hat{i}+4 \hat{k}

and

\overrightarrow{A C}=

\mathbf{5} \hat{\mathbf{i}}-\mathbf{2} \hat{\mathbf{j}}+\mathbf{4} \hat{\boldsymbol{k}}

are the sides of a triangle

A B C,

then the length of the median through

\boldsymbol{A}

is:

Q12MathsUnit 12: Vector Algebra

In a triangle

A B C,

right angled at the vertex

A,

if the position vectors

A, B

and

C

are respectively

3 \tilde{i}+\tilde{j}-\tilde{k},-\tilde{i}+

\mathbf{3} \tilde{j}+p \tilde{k}

and

5 \tilde{i}+q \tilde{j}-4 \tilde{k},

then the point

(p, q)

lies on a line

Q13MathsUnit 12: Vector Algebra

Given

\left|\overrightarrow{\boldsymbol{A}}_{1}\right|=2,\left|\overrightarrow{\boldsymbol{A}}_{2}\right|=\overrightarrow{\mathbf{3}}

and

\left|\vec{A}_{1}+\vec{A}_{2}\right|=3 .

Find the value of

\left(\overrightarrow{\boldsymbol{A}}_{1}+\mathbf{2} \overrightarrow{\boldsymbol{A}}_{2}\right) \cdot\left(\boldsymbol{3} \overrightarrow{\boldsymbol{A}}_{1}-\boldsymbol{4} \boldsymbol{\vec { A }}_{2}\right)

Q14MathsUnit 12: Vector Algebra

Assertion In each of the three planes determined by two of the lines

O A, O B, O C

(

O

being the origin), a straight line is drawn through

O

perpendicular to the third line. The three lines so determined are coplanar. Reason

(a \times b) \times c+(b \times c) \times a+(c \times a) \times

b=0,

where

O A=a, O B=b

and

\boldsymbol{O} \boldsymbol{C}=\boldsymbol{c}

Q15MathsUnit 12: Vector Algebra

Given that

A+B+C=0 .

Out of three vectors, two are equal in magnitude and the magnitude of third vector is

\sqrt{2}

times that of either of the two having equal magnitude. Then, the angles between the vectors are given by.

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