Question

2. Vectors u and v are given by: u=5i-4j+ pk v=2i+tj-3k (a) Express p in terms of t when u and v are perpendicular. ​

Answer 1

If u and v are perpendicular, then their dot product is zero:

$\vec u \cdot \vec v = (5\,\vec\imath-4\,\vec\jmath+p\,\vec k) \cdot (2\,\vec\imath+t\,\vec\jmath - 3\,\vec k) = 10 - 4t - 3p = 0$

Solve for p :

$10 - 4t - 3p = 0$

$3p = 10 - 4t$

$\boxed{p = \dfrac{10-4t}3}$