If a line makes angles 90°,135° and 45° with x,y and z-axes respectively, find its direction cosines.

Let direction cosines of the line l,m and n. l=cos90°=0 m=cos135°= \(-\frac{1}{\sqrt 2}\) n=cos45°= \(\frac{1}{\sqrt 2}\) Therefore, the direction cosines of the line are 0, \(-\frac{1}{\sqrt 2}\) and \(\frac{1}{\sqrt 2}\) .

If a line makes angle 90°,135°,45° with x,y and z-axes respectively,find its direction cosines.

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Let y ∈ R be such that the lines \(L_1:\frac{x+11}{1}=\frac{y+21}{2}=\frac{z+29}{3}\) and \(L_2:\frac{x+16}{3}=\frac{y+11}{2}=\frac{z+4}{\gamma}\) intersect. Let R₁ be the point of intersection of L₁ and L₂. Let O = (0, 0 ,0), and \(\hat{n}\) denote a unit normal vector to the plane containing both the lines L₁ and L₂.
Match each entry in List-I to the correct entry in List-II.

List - I		List - II
(P)	γ equals	(1)	\(-\hat{i}-\hat{j}+\hat{k}\)
(Q)	A possible choice for \(\hat{n}\) is	(2)	\(\sqrt{\frac{3}{2}}\)
(R)	\(\overrightarrow{OR_1}\) equals	(3)	1
(S)	A possible value of \(\overrightarrow{OR_1}.\hat{n}\) is	(4)	\(\frac{1}{\sqrt6}\hat{i}-\frac{2}{\sqrt6}\hat{j}+\frac{1}{\sqrt6}\hat{k}\)
		(5)	\(\sqrt{\frac{2}{3}}\)

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Notes on Three Dimensional Geometry

Concepts Used:

Vector Algebra

A vector is an object which has both magnitudes and direction. It is usually represented by an arrow which shows the direction(→) and its length shows the magnitude. The arrow which indicates the vector has an arrowhead and its opposite end is the tail. It is denoted as

The magnitude of the vector is represented as |V|. Two vectors are said to be equal if they have equal magnitudes and equal direction.

Vector Algebra Operations:

Arithmetic operations such as addition, subtraction, multiplication on vectors. However, in the case of multiplication, vectors have two terminologies, such as dot product and cross product.