How do you find the inner product and state whether the vectors are perpendicular given <7,-2,4>*<3,8,1>?

Jul 11, 2016

Dot product = 9. Vectors are not orthogonal.

Explanation:

$\vec{u} \cdot \vec{v} = {\sum}_{i = 1}^{n} {u}_{i} {v}_{i} \text{ for } \vec{u} , \vec{v} \in {\mathbb{R}}^{n}$

So, in ${\mathbb{R}}^{3}$ the dot product will be given by ${\sum}_{i = 1}^{3} {u}_{i} {v}_{i}$

$= 7 \cdot 3 + \left(- 2\right) \cdot 8 + 4 \cdot 1 = 9$

For orthogonality, think about the other definition of the dot product:

$\vec{u} \cdot \vec{v} = | \vec{u} | | \vec{v} | \cos \theta$

If the vectors are orthogonal, $\theta = \frac{\pi}{2}$ and the dot product is zero. This is how you can check orthogonality with the dot product. As you can see, the dot product of these two vectors is non-zero and thus they are not orthogonal.