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Question

Find the direction cosines of the vector joining the points A (1, 2, –3) and B (–1, –2, 1) directed from A to B.

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Solution

The given points are A( 1,2,3 )and B( 1,2,1 ).

Formula for vector equation when initial and terminal points are given is,

AB =( x 2 x 1 ) i ^ +( y 2 y 1 ) j ^ +( z 2 z 1 ) k ^

Here, x 1 =1, x 2 =1, y 1 =2, y 2 =2, z 1 =3 and z 2 =1.

Substitute all values in above formula, we get,

AB =( 11 ) i ^ +( 22 ) j ^ +( 1( 3 ) ) z ^ =2 i ^ 4 j ^ +4 z ^

Magnitude of vector AB is,

| AB |= ( 2 ) 2 + ( 4 ) 2 + 4 2 = 4+16+16 = 36 =6

Direction cosines of AB are ( x | AB | , y | AB | , z | AB | ).

Here, x=2, y=4 and z=4.

Substitute all values for direction cosines, we get,

( 2 6 , 4 6 , 4 6 )=( 1 3 , 2 3 , 2 3 )

Thus, direction cosines of AB are ( 1 3 , 2 3 , 2 3 ).


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