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Question
Find the direction cosines of the straight line passing through the points (5, 6, 7) and (7, 9, 13). Also, find the parametric form of vector equation and Cartesian equations of the straight line passing through two given points
Solution
Given points (5, 6, 7) and (7, 9, 13)
Direction ratios are (2, 3, 6)
So the straight line is parallel to `2hat"i" + 3hat"j" + 6hat"k"`
Direction cosines are `(x/"r", y/"r", z/"r")`
r = `sqrt(2^3 + 3^2 + 6^2)`
= `sqrt(49)`
= 7
Vector equation = `(2/7, 3/7, 6/7)`
`vec"r" = (5hat"i" + 6hat"j" + 7hat"k") + "t"(2hat"i" + 3hat"j" + 6hat"k")`
or
`vec"r" = (7hat"i" + 9hat"j" + 13hat"k") + "t"(2hat"i" + 3hat"j" + 6hat"k")`, t ∈ R
Cartesian equation
`(x - 5)/2 = (y - 6)/3 = (z - 7)/6`
or
`(x - 7)/2 = (y - 9)/3 = (z - 13)/6`
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