(i) A line OP through origin O is inclin...

(i) A line OP through origin O is inclined at `60^(@) and 30^(@)` to OY and OZ respectively . Find the angle at which it is inclined to OX (ii) What are the direction cosines of a line which of a line which is equally inclined to the axes ?

Text Solution

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`cos^(2) alpha + cos^(2)60^(@) + cos^(2) 30^(@) = 1`
`cos^(2) alpha + (1)/(4) + (3)/(4) = 1 `
`rArr cos alpha = 0`
`rArr alpha = 90^(@) `
(ii) Let ` alpha, beta , gamma` be the angles that a line makes with the coordinates axes , then
` cos^(2) alpha + cos^(2) beta + cos^(2) gamma = 1 `
Here ` alpha = beta = gamma`
`rArr 3cos^(2) alpha = 1 `
`rArr cos alpha pm (1)/(sqrt(3))`
Thus the required direction consines are `pm(1)/(sqrt(3)),pm(1)/(sqrt(3)),pm(1)/(sqrt(3))`

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The number of lines which are equally inclined to the axes is :

Find the direction cosines of a vector vecr which is equally inclined with OX,OY and OZ are

`+-1/(sqrt(3)),+-1/(sqrt(3)),+-1/(sqrt(3))`

`+-1/3,+-1/3,+-1/3`

`+-1/(sqrt(2)),+-1/2,+-1/2`

none of these

The direction cosines of a vector vecr which is equally inclined with OX,OY and OZ ar