Question

For the following vectors:
a = (1,-5,9), b = (-4,12,-6), c = (-3,5,-7)
Calculate the following dot products:
a) a·b
b) a·c
c) b·c

Answer 1

Answer on Question #52356 – Mathematics – Vector Calculus

Question

For the following vectors: $a = (1, -5, 9)$, $b = (-4, 12, -6)$, $c = (-3, 5, -7)$. Calculate the following dot products:

a) $a \cdot b$;

b) $a \cdot c$;

c) $b \cdot c$.

Solution

Let us write the formula for the dot (or scalar) product in terms of vector components. For the two three-dimensional vectors

$$\vec{a} = (a_1, a_2, a_3) = a_1 \vec{i} + a_2 \vec{j} + a_3 \vec{k}$$

and

$$\vec{b} = (b_1, b_2, b_3) = b_1 \vec{i} + b_2 \vec{j} + b_3 \vec{k},$$

where $\vec{i}, \vec{j}, \vec{k}$ – the standard unit vectors of orthogonal coordinate system, the dot product is defined as

$$(\vec{a} \cdot \vec{b}) = a_1 b_1 + a_2 b_2 + a_3 b_3.$$

Hence, using (1)-(3) for the given vectors we get:

a)

$$(\vec{a} \cdot \vec{b}) = 1 \cdot (-4) + (-5) \cdot 12 + 9 \cdot (-6) = -4 - 60 - 54 = -118,$$

b)

$$(\vec{a} \cdot \vec{c}) = 1 \cdot (-3) + (-5) \cdot 5 + 9 \cdot (-7) = -3 - 25 - 63 = -91,$$

c)

$$(\vec{b} \cdot \vec{c}) = (-4) \cdot (-3) + 12 \cdot 5 + (-6) \cdot (-7) = 12 + 60 + 42 = 114.$$

Answer: a) $(\vec{a} \cdot \vec{b}) = -118$; b) $(\vec{a} \cdot \vec{c}) = -91$; c) $(\vec{b} \cdot \vec{c}) = 114$.

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