Vectors in space

Japanese school year: Math C

What you learn

This section extends vector concepts into three-dimensional space, covering 3D coordinate components, dot products, and equations of geometric figures. These tools are indispensable for 3D computer graphics, physics simulations, and solid geometry. Prior proficiency in 2D vector arithmetic and dot products is strongly recommended.

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Key points

This formula calculates the magnitude (length) of a 3D vector $\vec{a}=(a_1, a_2, a_3)$ . Extending the Pythagorean theorem into three dimensions, it sums the squares of the three components $a_1, a_2, a_3$ and takes the square root.

|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}

This formula computes the dot product of two 3D vectors from their components. Multiply the corresponding coordinates together ( $a_1 b_1, a_2 b_2, a_3 b_3$ ) and take their sum.

\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3

This is the standard equation of a sphere in 3D space. It describes the locus of all points $(x, y, z)$ whose distance from the center $(a, b, c)$ equals the radius $r$ ( $r > 0$ ).

(x - a)^2 + (y - b)^2 + (z - c)^2 = r^2

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Practice 1Coming soon