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Question
Assertion (A): The acute angle between the line `barr = hati + hatj + 2hatk + λ(hati - hatj)` and the x-axis is `π/4`
Reason(R): The acute angle 𝜃 between the lines `barr = x_1hati + y_1hatj + z_1hatk + λ(a_1hati + b_1hatj + c_1hatk)` and `barr = x_2hati + y_2hatj + z_2hatk + μ(a_2hati + b_2hatj + c_2hatk)` is given by cosθ = `(|a_1a_2 + b_1b_2 + c_1c_2|)/sqrt(a_1^2 + b_1^2 + c_1^2 sqrt(a_2^2 + b_2^2 + c_2^2)`
Options
Both A and R are true and R is the correct explanation of A.
Both A and R are true but R is not the correct explanation of A.
A is true but R is false.
A is false but R is true.
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Solution
Both A and R are true and R is the correct explanation of A.
Explanation:
The equation of the x-axis may be written as `vecr = thati`. Hence, the acute angle θ between the given line and the x-axis is given by cosθ = `(|1 xx 1 + (-1) xx 0 + 0 xx 0|)/(sqrt(1^2 + (-1)^2 + 0^2) xx sqrt(1^2 + 0^2 + 0^2)) `
= `1/sqrt(2)`
⇒ θ = `π/4`
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