Pre-Calculus Vectors

Vectors: magnitude, direction, and component form

Q: How do you find the magnitude of a vector?

The magnitude equals the square root of a squared plus b squared, the Pythagorean length of its components.

Q: How do you find the direction of a vector?

The direction is arctangent of b over a, adjusted for the quadrant of the components.

20 practice questions 0 video lessons Theory + worked examples

Vectors: Magnitude, Direction, and Component Form

Texas Precalculus (TEKS) • Standard P.4(I) • Vectors

Vectors: Magnitude, Direction, and Component Form is the opening topic of Vectors in the Texas Essential Knowledge and Skills (Precalculus, §111.42). It is aligned to Standard P.4(I), which requires students to use vectors to represent magnitude and direction.

A vector has magnitude and direction, written \(\langle a,b\rangle\), with magnitude \(\sqrt{a^2+b^2}\) and direction \(\arctan\dfrac{b}{a}\).

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Theory

A vector has both magnitude (length) and direction. In component form it is written \(v=\langle a,b\rangle\), where \(a\) is the horizontal change and \(b\) the vertical.

Magnitude: \(\|v\|=\sqrt{a^2+b^2}\).
Direction: \(\theta=\arctan\dfrac{b}{a}\) (adjusted for quadrant).
Unit vector: \(\hat v=\dfrac{v}{\|v\|}\), magnitude 1 in the same direction.

From magnitude and direction to components: \(v=\langle \|v\|\cos\theta,\ \|v\|\sin\theta\rangle\).

\(v=\langle a,b\rangle\): components \(a\) and \(b\), magnitude \(\sqrt{a^2+b^2}\).

The core vector facts.

Magnitude, direction, and conversion:

\[\|v\|=\sqrt{a^2+b^2},\quad \theta=\arctan\dfrac{b}{a},\quad v=\langle \|v\|\cos\theta,\ \|v\|\sin\theta\rangle\]

magnitude is the square root of a squared plus b squared; components come from magnitude times cosine and sine of the direction

A unit vector scales \(v\) to length 1: \(\hat v=\dfrac{1}{\|v\|}\langle a,b\rangle\).

How to work with a vector

Magnitude: \(\sqrt{a^2+b^2}\).
Direction: \(\arctan\dfrac{b}{a}\), then fix the quadrant.
To components: multiply the magnitude by \(\cos\theta\) and \(\sin\theta\).
Unit vector: divide each component by the magnitude.

Example 1 — Component form from magnitude and direction

A vector has magnitude \(10\) at \(30^\circ\). Find its component form.

Solution

Components are \(\langle \|v\|\cos\theta,\ \|v\|\sin\theta\rangle\).

\(a\)	\(=\)	\(10\cos 30^\circ=5\sqrt3\)
\(b\)	\(=\)	\(10\sin 30^\circ=5\)
\(v\)	\(=\)	\(\langle 5\sqrt3,\ 5\rangle\)

component form is 5 root 3 comma 5

Example 2 — Magnitude

Find the magnitude of \(v=\langle 3,-4\rangle\).

Solution

Use \(\|v\|=\sqrt{a^2+b^2}\).

\(\\|v\\|\)	\(=\)	\(\sqrt{3^2+(-4)^2}\)
	\(=\)	\(\sqrt{25}=5\)

magnitude is 5

Example 3 — Direction angle

Find the direction of \(v=\langle 1,1\rangle\).

Solution

\(v\) is in Quadrant I, so \(\theta=\arctan\dfrac{1}{1}\).

\(\theta\)

\(=\)

\(\arctan 1=45^\circ\)

direction is 45 degrees

Example 4 — Unit vector

Find the unit vector in the direction of \(v=\langle 3,4\rangle\).

Solution

Divide the vector by its magnitude \(\|v\|=5\).

\(\hat v\)	\(=\)	\(\dfrac{1}{5}\langle 3,4\rangle\)
	\(=\)	\(\left\langle\dfrac{3}{5},\dfrac{4}{5}\right\rangle\)

unit vector is three fifths comma four fifths

Common pitfalls

Adjust the direction for the quadrant. \(\arctan\) alone can give the wrong angle.

Magnitude is never negative. It is a length.

A unit vector has magnitude 1, not the same length as \(v\).

Frequently asked questions

What is a vector?

A quantity with both magnitude and direction, written in component form \(\langle a,b\rangle\).

How do you find the magnitude of a vector?

\(\|v\|=\sqrt{a^2+b^2}\) — the Pythagorean length of its components.

How do you find the direction of a vector?

\(\theta=\arctan\dfrac{b}{a}\), adjusted for the quadrant of the components.

What is a unit vector?

A vector of magnitude 1 in a given direction, found by dividing a vector by its magnitude.

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Vector addition (component, parallelogram, end-to-end)

Vectors: magnitude, direction, and component form

Vectors: Magnitude, Direction, and Component Form

📖 Theory

How to work with a vector

Common pitfalls

Frequently asked questions

Theory