Problem 1. a) (\vec{OQ} = \hat{i} + \hat{j} + \hat{k}) or (\frac{1}{2}\hat{i} + \hat{j} + \frac{1}{2}\hat{k}) b) (cos~\theta = \frac{\vec{OQ} \cdot \vec{OR}}{|\vec{OQ}||\vec{OR}|} = \frac{\langle 1, 1, 1 \rangle \cdot \langle \frac{1}{2}, 1, \frac{1}{2} \rangle}{\sqrt{3}\sqrt{\frac{3}{2}}} = \frac{2\sqrt{2}}{3}) Problem 2. Velocity: (\vec{V} = \frac{d\vec{R}}{dt} = \langle -3sin(t), 3cos(t), 1 \rangle) Speed: (|\vec{V}| = \sqrt{9sin^2(t) + 9cos^2(t) + 1} = \sqrt{10}) Problem 3. a) Minors: [ \begin{bmatrix} 1 & 1 & 2 \ -2 & -2 & -2 \ -3 & -5 & -6 \end{bmatrix} ] Cofactors: [ \begin{bmatrix} 1 & -1 & 2 \ 2 & -2 & 2 \ -3 & 5 & -6 \end{bmatrix} ] Inverse: [ \frac{1}{2} \begin{bmatrix} 1 & 2 & -3 \ 2 & -2 & 2 \ -3 & 5 & -6 \end{bmatrix} ] b) (X = A^{-1}B = \begin{bmatrix} -3 \ 4 \ -4 \end{bmatrix}) Problem 4. Q = top of the ladder: (\vec{OQ} = \langle 0, Lsin(\theta) \rangle) R = bottom of the ladder: (\vec{OR} = \langle -Lcos(\theta), 0 \rangle) Midpoint: (\vec{OP} = \frac{1}{2}(\vec{OQ} + \vec{OR}) = \langle -\frac{L}{2}cos(\theta), \frac{L}{2}sin(\theta) \rangle) Parametric equations: (x = -\frac{L}{2}cos(\theta), y = \frac{L}{2}sin(\theta)) Problem 5. a) (\vec{P_0P_1} \times \vec{P_0P_2} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ -1 & -1 & 1 \ 0 & -2 & 1 \end{vmatrix} = \hat{i} + \hat{j} + 2\hat{k}). Area = (\frac{1}{2}|\vec{P_0P_1} \times \vec{P_0P_2}| = \frac{1}{2}\sqrt{6}) b) Normal vector: (\vec{P_0P_1} \times \vec{P_0P_2} = \hat{i} + \hat{j} + 2\hat{k}). Equation: (x + y + 2z = 3) c) Parametric equations for the line: (x = -1 + t, y = t, z = t) Substituting: (-1 + 4t = 3, t = 1), intersection point (0, 1, 1) Problem 6. a) (\frac{d}{dt}(\vec{R} \cdot \vec{R}) = \vec{V} \cdot \vec{R} + \vec{R} \cdot \vec{V} = 2\vec{R} \cdot \vec{V}) b) Assume |R| is constant; then (RR) = 2R.V = 0, i.e., R⊥V. c) Assume R = 0, so R is constant; then (\frac{d}{dt}(\vec{R} \cdot \vec{V}) = \vec{V} \cdot \vec{V} + \vec{R} \cdot \vec{A} = 0). Therefore, (\vec{R} \cdot \vec{A} = -|\vec{V}|^2)

18.02: Exam 1 Practice Solutions

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