https://file.aiquickdraw.com/mathbot/file/1725866361829fvfyvv0w.pdf

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.

Explore detailed solutions to problems involving vector algebra, matrix theory, and linear algebra. This includes finding vector projections, calculating dot products, and determining perpendicular components. Suitable for undergraduate students.

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