Question

Find a ⋅ b.

a = 10i + 9j, b = 4i + 3j

what do i have to do with the i

Answer 1

Answer on Question #42416 – Math – Analytic Geometry

Find a · b.

$$\mathrm{a} = 10\mathrm{i} + 9\mathrm{j},\ \mathrm{b} = 4\mathrm{i} + 3\mathrm{j}$$

what do i have to do with the i.

Solution

$\vec{i}$ and $\vec{j}$ are the unit vectors of the $X$ and $Y$ axes. They are perpendicular. So,

$$\vec{i} \cdot \vec{i} = 1,\ \vec{j} \cdot \vec{j} = 1,\ \vec{i} \cdot \vec{j} = \vec{j} \cdot \vec{i} = 0.$$

The scalar product of the vectors $\vec{a}$ and $\vec{b}$ is

$$\begin{array}{l} \vec{a} \cdot \vec{b} = \left(10\vec{i} + 9\vec{j}\right) \cdot \left(4\vec{i} + 3\vec{j}\right) = 10 \cdot 4 \cdot \left(\vec{i} \cdot \vec{i}\right) + 10 \cdot 3 \cdot \left(\vec{i} \cdot \vec{j}\right) + 9 \cdot 4 \cdot \left(\vec{j} \cdot \vec{i}\right) + 9 \cdot 3 \cdot \left(\vec{j} \cdot \vec{j}\right) = \\ = 40 \cdot 1 + 30 \cdot 0 + 36 \cdot 0 + 27 \cdot 1 = 67. \end{array}$$

Answer: 67.

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