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

Common Core Pre-Calculus • Standard N-VM.1 • Vectors

Vectors: Magnitude, Direction, and Component Form is the opening topic of Vectors in the Common Core State Standards. It is aligned to Standard N-VM.1, which requires students to recognize vectors and determine their 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