Maths · Scalar and vector products
In a triangle right angled at the vertex if the position vectors and are respect
In a triangle \( A B C, \) right angled at the vertex \( A, \) if the position vectors \( A, B \) and \( C \) are respectively \( 3 \tilde{i}+\tilde{j}-\tilde{k},-\tilde{i}+ \) \( \mathbf{3} \tilde{j}+p \tilde{k} \) and \( 5 \tilde{i}+q \tilde{j}-4 \tilde{k}, \) then the point \( (p, q) \) lies on a line
- A. makeing an obtuse angle with the positive direction of \( x- \) axis
- B. Parallel to \( x- \) axis
- C. Parallel to \( y- \) axis
- D. making an acute angle with the positive direction of \( x-a x i s \)
Step-by-step solution
Since triangle ABC is right-angled at A, vectors AB and AC are perpendicular. Compute AB = B - A = (-4, 2, p+1) and AC = C - A = (2, q-1, -3). Dot product AB·AC = 0 gives -8 + 2(q-1) -3(p+1) = 0, which simplifies to 2q - 3p = 13. This is a line with slope 3/2 > 0, so it makes an acute angle with the positive x-axis.
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