Getting You Unstuck.
0 like 0 dislike
a) Let \(\mathbf{v}:=(a, b, c)\) and \(\mathbf{x}:=(x, y, z)\) be any vectors in \(\mathbb{R}^{3}\). Viewed as column vectors, find a \(3 \times 3\) matrix \(A_{\mathrm{v}}\) so that the cross product \(\mathbf{v} \times \mathbf{x}=A_{\mathrm{v}} \mathbf{x}\).
\mathbf{v} \times \mathbf{x}=A_{\mathbf{v}} \mathbf{x}=\left(\begin{array}{ccc}
0 & -c & b \\
c & 0 & -a \\
-b & a & 0
x \\
y \\
where the anti-symmetric matrix \(A_{\mathrm{v}}\) is defined by the above formula.
b) From this, one has \(\mathbf{v} \times(\mathbf{v} \times \mathbf{x})=A_{\mathbf{v}}(\mathbf{v} \times \mathbf{x})=A_{\mathbf{v}}^{2} \mathbf{x}\) (why?). Combined with the cross product identity \(\mathbf{u} \times(\mathbf{v} \times \mathbf{w})=\langle\mathbf{u}, \mathbf{w}\rangle \mathbf{v}-\langle\mathbf{u}, \mathbf{v}\rangle \mathbf{w}\), show that
A_{\mathrm{v}}^{2} \mathrm{x}=\langle\mathrm{v}, \mathrm{x}\rangle \mathrm{v}-\|\mathrm{v}\|^{2} \mathrm{x}
c) If \(\mathbf{n}=(a, b, c)\) is a unit vector, use this formula to show that (perhaps surprisingly) the orthogonal projection of \(\mathbf{x}\) into the plane perpendicular to \(\mathbf{n}\) is given by
\mathbf{x}-(\mathbf{x} \cdot \mathbf{n}) \mathbf{n}=-A_{\mathbf{n}}^{2} \mathbf{x}=-\left(\begin{array}{ccc}
-b^{2}-c^{2} & a b & a c \\
a b & -a^{2}-c^{2} & b c \\
a c & b c & -a^{2}-b^{2}
\end{array}\right) \mathbf{x}
in Mathematics by Platinum (119,106 points) | 61 views

Related questions

0 like 0 dislike
1 answer
asked Jan 13 in Mathematics by MathsGee Platinum (119,106 points) | 61 views
0 like 0 dislike
0 answers

Q&A | Subjects | Request Private Tutor | eBook

Join the MathsGee Support Club where you get study and financial support for success from our community. LEARN - CONNECT - EARN

On the MathsGee Support Club, you can:

1. Ask questions

2. Answer questions

3. Vote on Questions and Answers

4. Tip your favourite community member(s)

5. Create Live Video Tutorials (Paid/Free)

6. Join Live Video Tutorials (Paid/Free)

7. Earn points for participating

Posting on the MathsGee Support Club

1. Remember the human

2. Behave like you would in real life

3. Look for the original source of content

4. Search for duplicates before posting

5. Read the community's rules


1. Answers to questions will be posted immediately after moderation

2. Questions will be queued for posting immediately after moderation

3. Depending on how many posts we receive, you could be waiting up to 24 hours for your post to appear. But, please be patient as posts will appear after they pass our moderation.

Q&A | Subjects | Request Private Tutor | eBook

MathsGee Tools

Math Worksheet Generator

Math Algebra Solver

Trigonometry Simulations

Vectors Simulations

Matrix Arithmetic Simulations

Matrix Transformations Simulations

Quadratic Equations Simulations

Probability & Statistics Simulations

PHET Simulations

Visual Statistics

Q&A | Subjects |Request Private Tutor | eBook