Find the measure of a acute angle betwee...

Find the measure of a acute angle between the line direction ratios are 5, 12, -13 and 3, -4, 5.

Text Solution

Verified by Experts

Let `theta` be the required acute angle between the lines.
`therefore cos theta = |(a_(1)a_(2)+b_(1)b_(2)+c_(1)c_(2))/(sqrt(a_(1)^(2)+b_(1)^(2)+c_(1)^(2))*sqrt(a_(2)^(2)+b_(2)^(2)+c_(2)^(2)))|`
Where `a_(1)=5, b_(1) = 12, c_(1) = -13 anda_(2)=3, b_(2) = -4, c_(2)=5`
`therefore cos theta = |(5(3)+12(-4)+(-13)(5))/(sqrt(5^(2)+(12)^(2)+(-13)^(2))*sqrt(3^(2)+(-4)^(2)+5^(2)))|`
` = |(15-48-65)/(sqrt(25+144+169)*sqrt(9+16+25))|=|(-98)/(13sqrt(2)xx5sqrt(2))|=(98)/(130)`
` therefore cos theta = (49)/(65)" "therefore cos ^(-1) ((49)/(65)).`

Topper's Solved these Questions

THREE DIMENSIONAL GEOMETRY
NAVNEET PUBLICATION - MAHARASHTRA BOARD|Exercise Theory Question|2 Videos
THREE DIMENSIONAL GEOMETRY
NAVNEET PUBLICATION - MAHARASHTRA BOARD|Exercise Example for Practice|14 Videos
QUESTION BANK 2021
NAVNEET PUBLICATION - MAHARASHTRA BOARD|Exercise BINOMIAL DISTRIBUTION|20 Videos
TRIGONOMETRIC FUNCTIONS
NAVNEET PUBLICATION - MAHARASHTRA BOARD|Exercise Multiple Choice Questions|12 Videos

The acute angle between the lines whose direction ratios are 1, 2, 2 and -3, 6, -2 is

`cos^(-1)((13)/(21))`

`cos^(-1)((5)/(21))`

`cos^(-1)((11)/(21))`

`cos^(-1)((5)/(9))`

The acute angle between the lines whose direction ratios are 3, 2, 6 and -2, 1, 2 is

`cos^(-1)((8)/(21))`

`cos^(-1)((12)/(21))`

`cos^(-1)((2)/(21))`

`cos^(-1)((2)/(3))`

The acute angle between the lines having direction ratios 5,4,1 and -3,2,1 is

`cos^(-1)(sqrt(3)/(6))`

`cos^(-1)((5)/(sqrt(3)))`

`cos^(-1)((sqrt(3))/(7))`

`cos^(-1)((sqrt(7))/(3))`