Vectors in Two Dimensions

A vector records both how far and in which direction; components turn that geometric idea into algebra.

Scalars and vectors

A scalar has magnitude only, such as temperature, mass, or time. A vector has magnitude and direction, such as displacement, velocity, or force.

Two vectors are equal when they have the same magnitude and direction, even if their arrows are drawn from different starting points.

Component notation

A vector in the plane can be written

or $v_x\mathbf i+v_y\mathbf j$. The components describe horizontal and vertical change.

The zero vector $\langle0,0\rangle$ has no direction and magnitude zero.

Magnitude

The length of $\vec v=\langle a,b\rangle$ is

This is the Pythagorean theorem applied to the component triangle. Magnitude is always nonnegative.

Direction angle

For a nonzero vector, a standard direction angle $\theta$ is measured counterclockwise from the positive $x$-axis. Since

inverse tangent gives a reference value, but the component signs determine the correct quadrant. A calculator's two-argument atan2 function handles this automatically when available.

Worked example: magnitude and direction

The signs, not the calculator's principal tangent output alone, locate the direction.

Vector addition and subtraction

Add or subtract corresponding components:

Geometrically, addition can be drawn head-to-tail or by the parallelogram rule. Subtraction means adding the opposite vector:

Scalar multiplication

For scalar $k$,

If $|k|>1$, magnitude grows; if $0<|k|<1$, it shrinks. A negative scalar reverses direction. The new magnitude is $|k|\|\vec v\|$.

Position and displacement vectors

The vector from point $A(x_1,y_1)$ to point $B(x_2,y_2)$ is

Order matters: $\overrightarrow{BA}=-\overrightarrow{AB}$. The distance $AB$ is the magnitude of the displacement vector.

Unit vectors

A unit vector has magnitude $1$. For nonzero $\vec v$,

It preserves direction while removing scale. A vector of magnitude $M$ in direction $\hat v$ is $M\hat v$.

Resolving a magnitude into components

A vector of magnitude $M$ at direction angle $\theta$ has components

Cosine supplies the horizontal component and sine the vertical component. Quadrant signs are built into the trig values.

Applications and resultants

Forces, velocities, and displacements combine through vector addition. Add all component vectors to find the resultant, then calculate its magnitude and direction.

Keep units consistent and distinguish a vector result from its magnitude alone.

Common mistakes

Adding magnitudes instead of components. Direction affects the resultant.

Reversing endpoint subtraction. For $\overrightarrow{AB}$, use $B-A$.

Using inverse tangent without a quadrant check. Component signs determine direction.

Forgetting absolute value in scaled magnitude. Magnitudes cannot be negative.

Calling a component pair a point without context. Points locate; vectors displace.

Quick self-check

• Are the horizontal and vertical components identified with signs?
• Does the Pythagorean magnitude match the drawing?
• Is the direction angle in the correct quadrant?
• Were vectors combined component by component?
• Does endpoint order match the requested displacement?
• Are the final magnitude, direction, and units all reported?

Related topics

• Vectors in Three Dimensions
• Dot Product
• Right Triangle Trigonometry
• Coordinate Plane