Math 243: Calculus II |
1. Find the lengths of the sides of the triangle with vertices
\(P(1,-3,-3)\), \(Q(7,0,3)\), and \(R(10,-6,-3)\), and determine if the triangle
is a right triangle.
2. Find an equation of the sphere that passes through the origin and
whose center is \((2,2,-1)\).
3. Write the equation of the sphere in standard form.
\[ x^2 + y^2 + z^2 + 2x - 6y - 4z = 22\]
4. Find the component form of the vector \(\mathbf{a} = \overline{AB}\)
where \(A(0,4,1)\) and \(B(3,4,-3)\).
5. Consider the vectors \(\mathbf{x} = \langle 1,3,-2 \rangle\) and
\(\mathbf{y} = \langle 0, -1, 8 \rangle\) in \(\mathbb{R}^3\). Find
a.) \(\mathbf{x} + \mathbf{y}\)
b.) \(\mathbf{y} - \mathbf{x}\)
c.) \(3\mathbf{x} - 2\mathbf{y}\)
d.) \(\mathbf{x}\cdot\mathbf{y}\)
e.) \(\mathbf{x}\times\mathbf{y}\)
6. Find a unit vector in the same direction as the vector
\(\mathbf{x} = -4\mathbf{i} + 3\mathbf{j} - \mathbf{k}\).
7. Find the vector that has the same direction as \(\langle 6, 9, -2
\rangle\) but has length \(2\).
8. Suppose \(A, B, C\) are vertices of a triangle. Find
\[ \overline{AB} + \overline{BC} + \overline{CA}.\]
9. Find \(\mathbf{a} \cdot \mathbf{b}\) if \(\Vert\mathbf{a}\Vert =
40\), \(\Vert\mathbf{b}\Vert = 90\), and \(\theta = 3\pi/4 \).
10. Find the angle between the vectors in \(\mathbb{R}^3\).
\[ \mathbf{a} = \langle 1, -4, 1 \rangle\ \ \ \ \mathbf{b} =
\langle 0, 6, -6 \rangle.\]
11. Find the scalar and vector projections of \(\mathbf{b}\) onto
\(\mathbf{a}\).
\[\mathbf{a} = \langle 6, 7, -6 \rangle\ \ \ \mathbf{b} = \langle
5, -1 ,1 \rangle\]
12. Prove the Parallelogram Law,
\[ \Vert \mathbf{a} + \mathbf{b}\Vert^2 + \Vert \mathbf{a} - \mathbf{b}\Vert^2
= 2\Vert\mathbf{a}\Vert^2 + 2\Vert\mathbf{b}\Vert^2.\]
13. Compute the cross product \(\mathbf{a} \times \mathbf{b}\) and show
that both \(\mathbf{a}\) and \(\mathbf{b}\) are orthogonal to the result.
\[\mathbf{a} = \langle 6, 7, -6 \rangle\ \ \ \mathbf{b} = \langle
5, -1 ,1 \rangle\]
14. Find the area of the parallelogram with vertices
\(A(-3,0)\), \(B(-1,4)\), \(C(6,3)\), and \(D(4,-1)\).
15. Use the scalar triple product to compute the volume of the
parallelogram determinged by the vectors
\(\langle 1, 3, 2 \rangle\), \(\langle 3, -3, 6\rangle\), and
\(\langle -2, 0, 1\rangle\).
16. Find the distance between the point \(P(1,1,1)\) and the line
\(\mathbf{r}(t) = \langle 3t, 1 - 2t, 2 + t \rangle\).
17. Find all three equations of the line through the points
\(P(0,\tfrac{1}{2}, 1)\) and \(Q(3,1,-4)\).
18. Determine whether the lines \(\ell_1\) and \(\ell_2\) are parallel,
skew, or intersecting. If intersecting, find the angle of intersection.
\[\begin{cases}
\mathbf{r}_1(t) = \langle 12+8t, 16-4t, 4 + 12t\rangle, \\
\mathbf{r}_2(s) = \langle 1+4s,3-2s,4+5s \rangle
\end{cases}\]
19. Find an equation of the plane through the points \(P(1,2,3)\),
\(Q(4,0,-1)\), and \(R(2,-4,-2)\).
20. Find the distance between the point and the given plane.
\[\begin{cases}
P(1,-3,2) \\
\Pi:\ \ 3x + 2y + 6z = 5
\end{cases}\]
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