Vector Basics: Scalar Multiplication, Unit Vectors, and Angle Between Vectors

Scalar multiplication and repeated addition

• Multiplying a scalar by a number is repeated addition: $a\times b$ equals adding $a$ to itself $b$ times (for integer $b$).
• This idea generalizes to vectors as scalar multiplication.

Vector scalar multiplication

• For a vector $\mathbf{a}$ and scalar $s$: $\mathbf{b} = s\mathbf{a}$.
• Magnitude scales by $|s|$; direction stays the same if s>0, opposite if s<0.
• Example: $2\mathbf{a} = \mathbf{a} + \mathbf{a}$.

Newton's second law and dimensions

• Vector form: $\mathbf{F} = m\mathbf{a}$, where $\mathbf{F}$ and $\mathbf{a}$ are vectors and $m$ is a scalar.
• Units: $[F] = [m][a] = \mathrm{kg} \cdot \mathrm{m}/\mathrm{s}^2 = \mathrm{N}$ (Newton).
• Example: mass $m=2\ \mathrm{kg}$ and acceleration magnitude $|\mathbf{a}|=5\ \mathrm{m}/\mathrm{s}^2$ gives magnitude $|\mathbf{F}| = m|\mathbf{a}| = 10\ \mathrm{N}$ in the direction of $\mathbf{a}$.

Vector components and unit vectors

• Any vector can be written in component form: $\mathbf{a} = a<em>x \hat{\mathbf{i}} + a</em>y \hat{\mathbf{j}} + a_z \hat{\mathbf{k}}$.
• Unit vectors: $\hat{\mathbf{i}}$ (x-direction), $\hat{\mathbf{j}}$ (y-direction), $\hat{\mathbf{k}}$ (z-direction).
• Components are scalars (with units); unit vectors carry no units.

2D/3D coordinate system and right-handed basis

• In 3D, a right-handed coordinate system means rotating from +x to +y aligns with +z (via the right-hand rule).

Example vectors in unit-vector form

• Example: $\mathbf{\mathbf{d}} = -3\hat{\mathbf{i}} - 4\hat{\mathbf{j}}$; \mathbf{a} = -3\hat{\mathbf{i}}.
• Components can be negative (left/down/negative direction).

Angle between vectors: methods

• Direction angle of a vector \mathbf{v} = vx\hat{\mathbf{i}} + vy\hat{\mathbf{j}}$is given by$\theta = \operatorname{atan2}(vy, vx) to place the angle in the correct quadrant.
• Angle between vectors can be found by:
  • Dot product: \cos\theta = \frac{\mathbf{a}\cdot\mathbf{d}}{|\mathbf{a}|\,|\mathbf{d}|},\quad \theta = \arccos\left(\frac{\mathbf{a}\cdot\mathbf{d}}{|\mathbf{a}|\,|\mathbf{d}|}\right).
  • Or via directional angles: \theta = |\thetaa - \thetad| with quadrant corrections.
• For numerical stability, prefer the dot-product method to avoid quadrant issues.

Worked example (xy-components): angle between (\mathbf{d}) and (\mathbf{a})

• Given: \mathbf{d} = -3\hat{\mathbf{i}} - 4\hat{\mathbf{j}},\quad \mathbf{a} = -3\hat{\mathbf{i}}.
• Magnitudes: |\mathbf{d}| = \sqrt{(-3)^2 + (-4)^2} = 5,\quad |\mathbf{a}| = \sqrt{(-3)^2} = 3.
• Dot product: \mathbf{d}\cdot\mathbf{a} = (-3)(-3) + (-4)(0) = 9.
• Cosine of angle: \cos\theta = \frac{9}{5\cdot 3} = 0.6.
• Angle: \theta = \arccos(0.6) \approx 53.13^{\circ}.
• Therefore, the angle between the vectors is approximately 53.1^{\circ}.

Quick directional note

• If you compute individual direction angles: \thetad = \operatorname{atan2}(-4,-3) \approx 233.13^{\circ},\quad \thetaa = \operatorname{atan2}(0,-3) = 180^{\circ}. The smaller angle is (|\thetad - \thetaa| \approx 53.13^{\circ}).

Key takeaways

• Scalar multiplication scales magnitude and may flip direction if negative.
• Vectors decompose into unit directions; component form is aX + bY + cZ with unit vectors.
• Use dot product for the angle between vectors to avoid quadrant errors; use \operatorname{atan2}$$ for individual vector directions when needed.