Math101 · Independent Ontario learning library
Math101Vectors in Two Dimensions
Two-dimensional vectors represent magnitude and direction through components, geometric arrows, and linear combinations.
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.
Worked example: magnitude and direction
The signs, not the calculator's principal tangent output alone, locate the direction.
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.
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.
