Vector Addition & Operations Visualizer

The grid runs from −10 to 10 on both axes. Vector $\vec a$ is a solid teal arrow and $\vec b$ a solid coral arrow, both starting at the origin. A dashed coral copy of $\vec b$ starts at the tip of $\vec a$ to show tip-to-tail addition, the sum $\vec a + \vec b$ is a thick navy arrow from the origin, and the scalar multiple $c\vec a$ is a dashed purple arrow.

The readout lists $\vec a$ and $\vec b$ with their magnitudes, the sum $\vec a + \vec b$ in component form, $|\vec a + \vec b|$ worked out as a square root, the direction of the sum as atan2(y, x) in degrees, and $c\vec a$ in component form. All readouts are rounded to one decimal place, and that includes c itself: a setting of 0.25 is displayed as 0.3, so for exact work stick to values of c that are whole numbers or end in .5.

Two habits of the readout to know. First, the direction is reported between −180° and 180°, so a vector pointing into Quadrant III or IV gets a negative angle; add 360° to convert to the 0° to 360° convention. Second, with large components the sum can reach 16 in either coordinate and run past the edge of the grid; the readout is still correct.

For $\vec a = \langle a_1, a_2 \rangle$ and $\vec b = \langle b_1, b_2 \rangle$:

The simulator does not show it, but the same topic also uses the dot product $\vec a \cdot \vec b = a_1 b_1 + a_2 b_2$, which is 0 exactly when the vectors are perpendicular.

Let $\vec a = \langle 4, 2 \rangle$ and $\vec b = \langle -1, 5 \rangle$. Find $\vec a + \vec b$, its magnitude and direction, and $1.5\vec a$. These are the simulator's starting values.

Step 1: add components. $\vec a + \vec b = \langle 4 + (-1),\ 2 + 5 \rangle = \langle 3, 7 \rangle$.

Step 2: magnitude. $|\vec a + \vec b| = \sqrt{3^2 + 7^2} = \sqrt{58} \approx 7.62$.

Step 3: direction. Both components are positive, so the sum is in Quadrant I and $\theta = \tan^{-1}\frac{7}{3} \approx 66.8^\circ$.

Step 4: scalar multiple. $1.5\langle 4, 2 \rangle = \langle 6, 3 \rangle$, with magnitude $1.5\sqrt{20} \approx 6.71$, pointing the same way as $\vec a$.

Check it in the simulator: the readout shows a + b = ⟨3, 7⟩, |a + b| = √(3²+7²) = 7.6, direction(a+b) = atan2(7, 3) = 66.8°, and c·a = 1.5·⟨4, 2⟩ = ⟨6, 3⟩. The unit vector in the direction of the sum is $\frac{1}{\sqrt{58}}\langle 3, 7\rangle \approx \langle 0.394, 0.919 \rangle$.

Follow-up (Quadrant III): set $\vec a = \langle -1, -1 \rangle$ and $\vec b = \langle -2, -3 \rangle$. The sum is $\langle -3, -4 \rangle$ with magnitude 5. $\tan^{-1}\frac{-4}{-3} = \tan^{-1}\frac43 \approx 53.1^\circ$ is the reference angle, but the vector points down and left, so $\theta = 180^\circ + 53.1^\circ = 233.1^\circ$. The simulator reports −126.9°, which is the same direction measured clockwise: $-126.9^\circ + 360^\circ = 233.1^\circ$.

Vectors are in Unit 4, which is not assessed on the AP Precalculus exam itself; they matter for class assessments, AP Calculus BC and AP Physics. The errors that come up most often:

Although Topic 4.8 sits outside the exam's scope, vector components and the sum of vectors are the language of displacement, velocity and force in AP Physics 1, and of motion along parametric curves in AP Calculus BC.