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Question
Find the equation of the line in vector and in Cartesian form that passes through the point with position vector `2hati -hatj+4hatk` and is in the direction `hati + 2hatj - hatk`.
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Solution
The required line passes through the given vector point `vec(r_1) = 2hati - hatj + 4hatk` and is parallel to the vector `vecb = hati + 2hatj - hatk`.
∴ Equation of required line `vecr = vec(r_1) + λ vecb`
or `vecr = (2hati - hatj + 4hatk) + λ(hati + 2hatj + hatk)` ........(i)
Cartesian Equations :
Taking `vecr = xhati + yhatj + zhatk` in equation (i),
`xhati + yhatj + zhatk = (2hati - hatj + 4hatk) + λ (hati + 2hatj - hatk)`
⇒ `xhati + yhatj + zhatk = (2 + λ)hati + (-1 + 2 λ)hatj + (4 - λ)hatk`
⇒ x = 2 + λ, y = −1 + 2λ, z = 4 − λ
⇒ `(x- 2)/1 = (y + 1)/2 = (z - 4)/-1 = λ`
∴ `(x- 2)/1 = (y + 1)/2 = (z - 4)/-1` is the cartesian equation of the line.
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