If a⃗ = -î - 2ĵ + 3k̂, b⃗ = 2î + 3ĵ - 5k̂, prove that a⃗ and a⃗ × b⃗ are perpendicular.

If $\vec{a} = -\hat{i} - 2\hat{j} + 3\hat{k}$, $\vec{b} = 2\hat{i} + 3\hat{j} - 5\hat{k}$, prove that $\vec{a}$ and $\vec{a} \times \vec{b}$ are perpendicular.

Given: $\vec{a} = -\hat{i} - 2\hat{j} + 3\hat{k}$, $\vec{b} = 2\hat{i} + 3\hat{j} - 5\hat{k}$

To show: $\vec{a}$ and $\vec{a} \times \vec{b}$ are perpendicular

Open in the 3D geometry and vectors calculator

Answer

$\vec a\times\vec b = \hat i+\hat j+\hat k$ and $\vec a\cdot(\vec a\times\vec b) = 0$, so $\vec a$ and $\vec a\times\vec b$ are perpendicular.

1. $\vec a = -\hat i-2\hat j+3\hat k$, $\vec b = 2\hat i+3\hat j-5\hat k$ (given).
2. $\vec a\times\vec b = \begin{vmatrix}\hat i&\hat j&\hat k\\-1&-2&3\\2&3&-5\end{vmatrix}$ (definition of cross product).
3. $= \hat i(10-9)-\hat j(5-6)+\hat k(-3+4) = \hat i+\hat j+\hat k$ (expanding the determinant).
4. $\vec a\cdot(\vec a\times\vec b) = (-1)(1)+(-2)(1)+(3)(1)$ (dot product).
5. $= -1-2+3 = 0$.
6. Two non-zero vectors with zero dot product are perpendicular, so $\vec a$ and $\vec a\times\vec b$ are perpendicular.