Find the angle between two vectors when k = -9

Key takeaway: When two vectors are given by components, the cleanest route is usually the dot product. It lets you connect the algebra to the geometric angle without guessing.

Use the dot product formula

$a\cdot b = |a||b|\cos\theta$

When $k=-9$, substitute the value into the vector expressions for $a$ and $b$ from the earlier parts of the question. Then:

1. Find $a\cdot b$.
2. Find the magnitudes $|a|$ and $|b|$.
3. Rearrange to get $\cos\theta = \frac{a\cdot b}{|a||b|}$
4. Calculate $\theta = \cos^{-1}(\dots)$

For a 1-mark part, the expected answer is usually the angle in degrees rounded appropriately.

If your earlier expressions give a negative dot product, the angle will be obtuse; if the dot product is positive, the angle will be acute.

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Pitfalls the pros know 👇 A common mistake is using the raw components without first substituting $k=-9$ everywhere. Another easy slip is forgetting that the angle must come from the inverse cosine, not from the dot product alone.

What if the problem changes? If the value of $k$ changes, the same method still works: substitute the new $k$, compute the dot product and both magnitudes, then evaluate $\cos^{-1}\!\left(\frac{a\cdot b}{|a||b|}\right)$. A different $k$ may change the angle from acute to right or obtuse.

Tags: dot product, angle between vectors, vector magnitude