Let $$u = \langle 1, −2, 3 \rangle \qquad \text{and} \qquad v = \langle −4, 5, 6 \rangle.$$
Find the angle between $u$ and $v$, first by using the dot product and then using the cross product.
I used the formula:
$U \cdot V = ||u|| \, ||v|| \cos \Delta$
and got $83^\circ$ from the dot product.
However, I am lost as how to use the cross product to find the answer.
Best Answer
Hint The cross product satisfies $$||{\bf a} \times {\bf b}|| = ||{\bf a}|| \, ||{\bf b}|| \sin \theta,$$ where $\theta \in [0, \pi]$ is the angle between $\bf a$ and $\bf b$.
(In fact this property is very nearly one of the common definitions of the cross product; see, e.g., Defn. 7.4 of Dennis G. Zill, Michael R. Cullen (2006). Advanced engineering mathematics (3rd ed.). Jones & Bartlett Learning.)