Question

a and b are vectors defined by a = 8i + 2j - 3k and b = 3i - 6j + 4k, where I,j,k are mutually perpendicular unit vectors. Show that a and b are perpendicular to each other.

a. 90
b. 45
c. 1
d. 0

Answer 1

Answer on Question #51024 – Math – Analytic Geometry

$a$ and $b$ are vectors defined by $a = 8i + 2j - 3k$ and $b = 3i - 6j + 4k$, where $i, j, k$ are mutually perpendicular unit vectors. Show that $a$ and $b$ are perpendicular to each other.

Solution

To show that $a$ and $b$ are perpendicular vectors, we must find scalar (dot) product of these vectors.

$(a, b) = |a||b| \cos \alpha$, where $\alpha$ is angle between $a$ and $b$. If $a$ and $b$ are perpendicular, then $\alpha$ is equal to $\frac{\pi}{2}$. Then $(a, b) = 0$. So, in our case we have

$$\begin{array}{l} (a, b) = (8i + 2j - 3k, 3i - 6j + 4k) = 8 \cdot 3 + 2 \cdot (-6) + (-3) \cdot 4 = \\ = 24 - 12 - 12 = 0 \end{array}$$

Answer: $a$ and $b$ are perpendicular vectors.

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