PHSX 220 Physics I (w/ Calculus) · Unit 1: Kinematics & Newton’s laws · on Exam 1
Ch 2 Vectors: components and algebra
Study notes for Vectors: components and algebra in PHSX 220 Physics I (w/ Calculus) at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- A vector has magnitude and direction; a scalar has only size. Displacement, velocity, acceleration, force and momentum are vectors. Distance, speed, mass, energy and time are scalars.
- Components turn vector problems into two ordinary problems: $A_x = A\cos\theta$, $A_y = A\sin\theta$ with $\theta$ measured counterclockwise from the $+x$ axis. Going back: $A = \sqrt{A_x^2 + A_y^2}$, $\tan\theta = A_y/A_x$ (check the quadrant from the signs of the components).
- Add vectors by adding components: $\vec R = \vec A + \vec B \Rightarrow R_x = A_x + B_x,\ R_y = A_y + B_y$. Subtracting $\vec B$ means adding $-\vec B$ (flip both components).
- Unit vectors $\hat i, \hat j, \hat k$ point along $x, y, z$: $\vec A = A_x\hat i + A_y\hat j$. Multiplying by a scalar scales the magnitude (negative scalar reverses direction).
- Dot product $\vec A\cdot\vec B = AB\cos\theta = A_xB_x + A_yB_y$ (a scalar; used for work). Cross product $|\vec A\times\vec B| = AB\sin\theta$ (a vector perpendicular to both; used for torque and angular momentum; right-hand rule).
Key formulas
Components
$$A_x = A\cos\theta,\quad A_y = A\sin\theta,\quad A = \sqrt{A_x^2+A_y^2},\quad \theta = \tan^{-1}\!\frac{A_y}{A_x}$$
Dot and cross
$$\vec A\cdot\vec B = AB\cos\theta = A_xB_x + A_yB_y,\qquad |\vec A\times\vec B| = AB\sin\theta$$
Worked example
$\vec A$ has magnitude 5.0 m at $37^\circ$ above the $+x$ axis; $\vec B$ is 8.0 m along $-y$. Find $\vec A + \vec B$.
Show the solution
$A_x = 5\cos37^\circ = 3.99$, $A_y = 5\sin37^\circ = 3.01$; $B_x = 0$, $B_y = -8$. $R_x = 3.99$, $R_y = -4.99$. $R = \sqrt{3.99^2 + 4.99^2} = 6.39$ m at $\tan^{-1}(-4.99/3.99) = -51.3^\circ$, i.e. $51.3^\circ$ below the $+x$ axis.
Common mistakes
- Using $\cos$ for the $y$ component. Only true if the angle is measured from the $y$ axis: draw the triangle every time.
- Reporting the calculator’s arctan without checking the quadrant (it never returns angles in quadrants II or III).
- Adding magnitudes instead of components.
Exam tipEvery 2D force or motion problem starts the same way: draw axes, resolve every vector into components, then work each axis separately.
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