Vector Components and Addition – Quick Notes

Vector components: magnitude and direction

  • A vector is described by magnitude and direction: magnitude ff and angle θ\theta from a reference axis.

  • Decompose a vector into components:

    • fx=fcos⁡θf_x = f \cos\theta

    • fy=fsin⁡θf_y = f \sin\theta

  • Reconstruct: f=⟨fx,fy⟩=fxi^+fyj^\mathbf{f} = \langle f_x, f_y \rangle = f_x \hat{i} + f_y \hat{j}

From magnitude and direction to components

  • Given ff and θ\theta: the components are fx=fcos⁡θ,fy=fsin⁡θf_x = f \cos\theta, \quad f_y = f \sin\theta

From components to magnitude and direction

  • Magnitude: f=fx2+fy2f = \sqrt{f_x^2 + f_y^2}

  • Direction: θ=tan⁡−1(fyfx)\theta = \tan^{-1}\left( \frac{f_y}{f_x} \right)

  • Alternative relationships: cos⁡θ=fxf,sin⁡θ=fyf\cos\theta = \frac{f_x}{f}, \quad \sin\theta = \frac{f_y}{f}

Vector addition by components (tip-to-tail)

  • For vectors a\mathbf{a} and b\mathbf{b}, resultant r=a+b\mathbf{r} = \mathbf{a} + \mathbf{b}.

  • Component-wise addition:

    • rx=ax+bxr_x = a_x + b_x

    • ry=ay+byr_y = a_y + b_y

  • Subtraction as addition of negative:

    • dx=ax−bxd_x = a_x - b_x

    • dy=ay−byd_y = a_y - b_y

  • Magnitude/angle of the result: use r=rx2+ry2,θr=tan⁡−1(ryrx)r = \sqrt{r_x^2 + r_y^2}, \quad \theta_r = \tan^{-1}\left(\frac{r_y}{r_x}\right)

Example (numbers from transcript)

  • Vector a\mathbf{a}: magnitude 22 at 60∘60^\circ; b\mathbf{b}: magnitude 11 at 30∘30^\circ.

  • Components:

    • ax=2cos⁡60∘=1.00a_x = 2\cos60^\circ = 1.00

    • ay=2sin⁡60∘=1.73a_y = 2\sin60^\circ = 1.73

    • bx=1cos⁡30∘=0.87b_x = 1\cos30^\circ = 0.87

    • by=1sin⁡30∘=0.50b_y = 1\sin30^\circ = 0.50

  • Resultant components:

    • rx=ax+bx=1.00+0.87=1.87r_x = a_x + b_x = 1.00 + 0.87 = 1.87

    • ry=ay+by=1.73+0.50=2.23r_y = a_y + b_y = 1.73 + 0.50 = 2.23

  • Resultant magnitude and angle:

    • r=1.872+2.232≈2.91r = \sqrt{1.87^2 + 2.23^2} \approx 2.91

    • θr=tan⁡−1(2.231.87)≈50∘\theta_r = \tan^{-1}\left(\frac{2.23}{1.87}\right) \approx 50^\circ

Quick reference formulas (summary)

  • Components from vector: fx=fcos⁡θ,fy=fsin⁡θf_x = f \cos\theta, \quad f_y = f \sin\theta

  • Vector from components: f=fx2+fy2,θ=tan⁡−1(fyfx)f = \sqrt{f_x^2 + f_y^2}, \quad \theta = \tan^{-1}\left(\frac{f_y}{f_x}\right)

  • Subtraction via components: Dx=Ax−Bx,Dy=Ay−ByD_x = A_x - B_x, \quad D_y = A_y - B_y

  • Addition via components: Rx=Ax+Bx,Ry=Ay+ByR_x = A_x + B_x, \quad R_y = A_y + B_y