Mathematics · JEE

Important Questions: Direction ratios and direction cosines and the angle between two intersecting lines for JEE

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Syllabus context

Part of Three Dimensional Geometry in JEE Main Mathematics.

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Q1MathsUnit 11: Three Dimensional Geometry

The plane

\boldsymbol{x}-\mathbf{2} \boldsymbol{y}+\boldsymbol{z}-\boldsymbol{6}=\mathbf{0}

and the line

\frac{x}{1}=\frac{y}{2}=\frac{z}{3}

are related as

Q2MathsUnit 11: Three Dimensional Geometry

\operatorname{Let} \boldsymbol{A}(\mathbf{2} \hat{\boldsymbol{i}}+\boldsymbol{3} \hat{\boldsymbol{j}}+\mathbf{5} \hat{\boldsymbol{k}}) \boldsymbol{B}(-\hat{\boldsymbol{i}}+\boldsymbol{3} \hat{\boldsymbol{j}}+2 \hat{\boldsymbol{k}})

and

C(\lambda \hat{i}+5 \hat{j}+\mu \hat{k})

are vertices of

a

triangle and its median through

A

is equally inclined to the positive directions of the axes. The value of

\lambda+

\mu

is equal to

Q3MathsUnit 11: Three Dimensional Geometry

The direction ratios of the joining

A(1,2,1)

and (2,1,2) are

Q4MathsUnit 11: Three Dimensional Geometry

The projection of the line segment joining (0,0,0) and (5,2,4) on the line whose direction ratios are 2,-3,6 is

Q5MathsUnit 11: Three Dimensional Geometry

The planes

2 x-y+4 z=5

and

5 x-

2.5 y+10 z=6

are

Q6MathsUnit 11: Three Dimensional Geometry

The number of straight lines that are equally inclined to the threedimensional coordinate axes, is

Q7MathsUnit 11: Three Dimensional Geometry

If a ray makes angles

\alpha, \beta, \gamma

and

\delta

with the four diagonals of a cube and

\mathbf{A}: \cos ^{2} \boldsymbol{\alpha}+\cos ^{2} \boldsymbol{\beta}+\cos ^{2} \boldsymbol{\gamma}+\cos ^{2} \boldsymbol{\delta}

\mathbf{B}: \sin ^{2} \boldsymbol{\alpha}+\sin ^{2} \boldsymbol{\beta}+\sin ^{2} \boldsymbol{\gamma}+\sin ^{2} \boldsymbol{\delta}

\mathbf{C}: \cos 2 \boldsymbol{\alpha}+\cos 2 \boldsymbol{\beta}+\cos 2 \gamma+\cos 2 \boldsymbol{\delta}

Arrange

A, B, C

in descending order

Q8MathsUnit 11: Three Dimensional Geometry

\theta

is the angle between the lines

A B, A C

where

A, B, C

are the three points with coordinates (1,2,-1),(2,0,3),(3,-1,2) respectively, then

\sqrt{462} \cos \theta

is equal to

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Key areas include Direction ratios and direction cosines and the angle between two intersecting lines. Prioritise these before moving to edge cases.

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