Homework Chapter 3 (being covered before Ch. 2) UTK – M251 – Matrix Algebra Fall 2003, Jochen Denzler, MWF 1:25–2:15, Ayres 318

1. Find the components of u⃗, v⃗, u⃗ + v⃗, and u⃗ − v⃗, for the vectors given in the following figure: The radius of the circle is 1; the angles shown are meant to be 45^{∘} and 150^{∘} (the figure is not quite to scale).

[figure: x; y; u⃗; v⃗]

2. Given the point F with coordinates (0, 0, 1). The following is known about the point P with coordinates (a, b, c): it has the same distance from F as it has from the xy plane. Write this information in form of an equation involving a, b, and c. You may need to think a bit, and reread Ch. 3.2. — FYI: Problem 13 on p. 129 is similar, but not quite the same. And if you need more help, you should come in.

3. Find a unit vector x⃗ = [x_{1}, x_{2}]^{T} in the plane that is orthogonal to w⃗ = [3, 4]^{T}. How many such unit vectors are there?

4. Find a unit vector x⃗ = [x_{1}, x_{2}, x_{3}]^{T} in space that is orthogonal to both u⃗ = [1, 0, 1]^{T} and v⃗ = [0, 1, 1]^{T}.

5. Given the vectors x⃗ = [x_{1}, x_{2}, x_{3}]^{T} and y⃗ = [y_{1}, y_{2}, y_{3}]^{T} in space, I have defined their cross product x⃗ × y⃗ to be the vector w⃗ = [x_{2}y_{3} − x_{3}y_{2} , x_{3}y_{1} − x_{1}y_{3} , x_{1}y_{2} − x_{2}y_{1}]^{T}. Show that indeed, ∥w⃗∥^{2} = ∥x⃗∥^{2}∥y⃗∥^{2}(1 − cos^{2} φ), where φ is the angle between x⃗ and y⃗.

6. With the same definitions as in the previous problem, check that w⃗ is orthogonal to x⃗.

7. Let u⃗ = [2, 1, 3]^{T}, v⃗ = [−1, 0, 4]^{T}, w⃗ = [2, −1, −3]^{T}. Calculate v⃗ × w⃗, u⃗ × (v⃗ × w⃗), u⃗ × v⃗, (u⃗ × v⃗) × w⃗.

8. With the vectors from the previous problem, calculate u⃗ · (v⃗ × w⃗) and v⃗ · (w⃗ × u⃗).

9. Find the area of the triangle whose vertices are the points A(1, 1, 3), B(−2, 3, 0), C(1, 1, −2).

10. Some of the following expressions are meaningless, for vectors u⃗, v⃗, w⃗ in space. Find them, and explain why they are meaningless. For those that are meaningful, specify whether the result is scalar or vector.

\begin{matrix}(a) & (u⃗ × v⃗) × w⃗ \\ (b) & u⃗ × v⃗ × w⃗ \\ (c) & (u⃗ · v⃗) × w⃗ \\ (d) & (u⃗ · v⃗)w⃗ \\ (e) & (u⃗ · v⃗) · w⃗ \\ (f) & (u⃗ × v⃗) · w⃗ \end{matrix}

M251  p. 1  Homework Ch. 3