The worksheet contains a series of vector algebra, matrix theory, and linear algebra problems. I'll go step by step for each question.

1. Given $A = (3,4,1,-1)$ and $B = (2,-1,2,-2)$:

a) $\text{Proj}_B(A)$: Projection of $A$ onto $B$
b) $\text{Proj}_{B^\perp}(A)$: Projection of $A$ onto the plane perpendicular to $B$
c) $\text{Proj}_{2B}(3A)$: Projection of $3A$ onto $2B$

To find projections, use the formula for the projection of vector $A$ onto vector $B$: $\text{Proj}_B(A) = \frac{A \cdot B}{B \cdot B} B$ I'll solve the first part as an example. For $A \cdot B$ and $B \cdot B$: $A \cdot B = (3)(2) + (4)(-1) + (1)(2) + (-1)(-2) = 6 - 4 + 2 + 2 = 6$ $B \cdot B = (2)^2 + (-1)^2 + (2)^2 + (-2)^2 = 4 + 1 + 4 + 4 = 13$ So, the projection is: $\text{Proj}_B(A) = \frac{6}{13} (2,-1,2,-2) = \left(\frac{12}{13}, -\frac{6}{13}, \frac{12}{13}, -\frac{12}{13}\right)$

b) For the perpendicular component, $\text{Proj}_{B^\perp}(A) = A - \text{Proj}_B(A)$.

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2. Given $a = i - j + k$ and $b = 3i + 4j + k$:

a) Find the unit vector in the direction of $a + b$. b) Find the vector opposite to $a \times b$ with a norm of 6.

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3. Let $v = i + j$. If $u = ai + aj + ck$ is a unit vector and $\| v - 2u \| = \sqrt{2}$, find $u$.

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4. Show that the vectors $2i - j + k$, $i - 3j - 5k$, and $3i - 4j - 4k$ form the sides of a right-angled triangle, and find the remaining angles.

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5. If vectors $a$ and $b$ make an angle $\theta = \frac{\pi}{4}$, with $\| a \| = \sqrt{2}$ and $\| b \| = 1$, find the cosine of the angle between $a + b$ and $a - b$.

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Continuing...

Each problem follows a specific concept from vector algebra, matrices, and systems of equations. I will provide further details for each question if requested.

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Would you like detailed solutions for more of the questions? Here are 5 questions that extend from this content:

1. What is the perpendicular component of vector $A$ onto vector $B$?
2. How do you find the angle between two vectors using the dot product?
3. How is the area of a triangle calculated using vectors?
4. What are the steps to find the volume of a parallelepiped using vector products?
5. How do you find the equation of a line parallel to a given line in 3D space?

Tip: For vector-related questions, always break down the components into dot products or cross products to simplify calculations.