Components & magnitude

Japanese school year: Math C

What you learn

In this section, you will learn to represent geometric vectors using coordinate components and calculate their magnitudes, additions, and scalar multiplications. This algebraic representation simplifies vector operations in physics and geometry. Basic understanding of geometric vectors and the Pythagorean theorem is recommended beforehand.

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Key points

This formula calculates the magnitude (length) of a 2D vector $\vec{a}=(a_1, a_2)$ . Applying the Pythagorean theorem, it squares the $x$ -component $a_1$ and $y$ -component $a_2$ , adds them together, and takes the square root.

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

This formula finds the components of vector $\vec{AB}$ directed from starting point $A(x_1, y_1)$ to endpoint $B(x_2, y_2)$ . It is calculated by subtracting the coordinates of $A$ component-wise from the coordinates of $B$ .

\vec{AB} = (x_2 - x_1, \, y_2 - y_1)

This formula defines the addition of two vectors in component form. You obtain the resultant vector by adding corresponding $x$ -components ( $a_1+b_1$ ) and $y$ -components ( $a_2+b_2$ ) separately.

\vec{a} + \vec{b} = (a_1 + b_1, \, a_2 + b_2)

This formula calculates scalar multiplication of a vector $\vec{a}$ by a real number $k$ . To scale the vector's length or reverse its direction, multiply each component $a_1$ and $a_2$ by $k$ .

k\vec{a} = (ka_1, \, ka_2)

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Practice 1Coming soon