Dot products

Japanese school year: Math C

What you learn

This section covers the definition of the dot product using vector magnitudes and angles, as well as its algebraic computation via components. It is essential for checking orthogonality and calculating mechanical work in physics. Prior familiarity with vector components, magnitudes, and cosine values from trigonometry is required.

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Key points

This fundamental formula defines the dot product of two vectors $\vec{a}$ and $\vec{b}$ . It multiplies their magnitudes $|\vec{a}|$ and $|\vec{b}|$ by the cosine of the angle $\theta$ between them.

\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta

This formula computes the dot product of two vectors directly from their components. By summing the products of matching components ( $a_1 b_1 + a_2 b_2$ ), it determines the dot product without needing the angle.

\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2

This property states that the dot product of any vector $\vec{a}$ with itself equals the square of its magnitude $|\vec{a}|^2$ . It is frequently used to determine vector lengths and distances.

\vec{a} \cdot \vec{a} = |\vec{a}|^2

This condition determines whether two non-zero vectors $\vec{a}$ and $\vec{b}$ are perpendicular. Because $\cos 90^\circ = 0$ , two vectors are orthogonal if and only if their dot product is $0$ .

\vec{a} \perp \vec{b} \iff \vec{a} \cdot \vec{b} = 0 \quad (\vec{a} \neq \vec{0}, \, \vec{b} \neq \vec{0})

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Practice 1Coming soon