Vectors in two dimensions are often easier to use when split into perpendicular parts. This page focuses on using right-triangle trigonometry and the Pythagorean theorem to move between a vector’s magnitude/angle and its components.

Why resolve vectors into components?

Many physics quantities are vectors (force, velocity, displacement). In 2D, you can treat one vector as two independent, perpendicular “shadow” vectors along chosen axes (often $x$ and $y$). This lets you apply algebra separately in each direction.

Components (what you are finding)

Components are not extra vectors added to the situation; they are just an equivalent representation of the same vector.

The right-triangle picture

Resolving a vector uses a right triangle:

A 2D vector is drawn as the hypotenuse of a right triangle, with the legs representing the perpendicular components along the coordinate axes. The labels show how the components relate to the magnitude via $A_x = A\cos\theta$ and $A_y = A\sin\theta$ when $\theta$ is measured from the +$x$ axis. This picture helps prevent the common sine/cosine swap by making “adjacent” and “opposite” explicit. Source

• The hypotenuse is the vector’s magnitude $A$.

• The legs are the component magnitudes $|A_x|$ and $|A_y|$.

• The angle $\theta$ is measured from one axis to the vector (you must state which axis you’re using).

Trig relationships for components

If $\theta$ is measured from the +$x$ axis toward the vector, then:

• The adjacent side corresponds to the $x$-component.

• The opposite side corresponds to the $y$-component.

Use cosine for adjacent/hypotenuse and sine for opposite/hypotenuse. The sign of each component comes from direction (left/down negative if your axes define them that way), not from the trig function alone.

A common mistake is swapping sine and cosine because the angle is actually measured from the $y$ axis. If $\theta$ is from +$y$, then the “adjacent” side is $|A_y|$, so the roles of sine/cosine switch.

Using the Pythagorean theorem appropriately

The Pythagorean theorem applies only when the components are perpendicular:

• Valid: $x$ and $y$ axes (90° apart)

• Not valid: two components chosen along non-perpendicular directions

When you compute $A=\sqrt{A_x^2+A_y^2}$, the squared terms remove signs; magnitude is always non-negative.

The vector and its perpendicular components are shown forming a right triangle, so the magnitude follows from the Pythagorean theorem: $A = \sqrt{A_x^2 + A_y^2}$. By tying the geometry to the algebra, the figure clarifies why this magnitude formula is valid specifically for orthogonal $x$–$y$ components. It also reinforces that component signs affect direction, not the nonnegative magnitude. Source

Direction must be handled separately.

Angle from components (direction)

To find direction from known components, the basic trig idea is:

• $\tan\theta = \dfrac{A_y}{A_x}$ (when $\theta$ is from +$x$)

However, the correct direction depends on the quadrant (signs of $A_x$ and $A_y$). Many calculators provide an “atan2” function (or equivalent) to determine the correct quadrant; otherwise, you must adjust the angle based on signs.

Checklist for AP Physics 1 Algebra problems

• Choose axes and define + directions clearly.

• Identify the angle’s reference axis (from +$x$ or +$y$).

• Use sine/cosine based on opposite/adjacent relative to that angle.

• Assign signs to $A_x$ and $A_y$ from direction.

• Use the Pythagorean theorem only for perpendicular components.