Solve for k when two vectors are perpendicular

Key concept: Perpendicular vectors are one of the nicest dot product applications: the calculation is short, and the condition is exact.

Step by step

For vectors to be perpendicular, their dot product must be zero:

$a\cdot b = 0$

So the method is:

1. Write out the component forms of $a$ and $b$.
2. Compute the dot product in terms of $k$.
3. Set the result equal to zero.
4. Solve the resulting equation for $k$.

That gives all values of $k$ for which the vectors are perpendicular.

If the dot product simplifies to a linear equation, you should get one value of $k$. If it becomes quadratic, there may be two solutions.

Pitfall alert

Don’t confuse perpendicular with parallel: perpendicular means $a\cdot b=0$, not equal slopes or matching components. Also, make sure every $k$ term is included before solving.

Try different conditions

If the question asks for a specific angle instead of perpendicularity, the dot product equation becomes $a\cdot b = |a||b|\cos\theta.$ For $90^\circ$, this still reduces to $a\cdot b=0$.

Further reading

orthogonal vectors, dot product, vector equation