Q1. Compute the magnitude of the vector .

Background

Topic: Vector Magnitude in

This question tests your understanding of how to compute the length (or norm) of a vector in three-dimensional space.

Key Terms and Formula:

• Magnitude (or norm) of a vector is given by:

1. Identify the components of : , , .

2. Square each component: , , and .

3. Add the squared values together.

4. Take the square root of the sum to find the magnitude.

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Q2. Find a unit vector in the direction of .

Background

Topic: Unit Vectors

This question asks you to find a vector with length 1 that points in the same direction as .

Key Terms and Formula:

• Unit vector:

• Magnitude formula as above.

1. Compute the magnitude of using .

2. Divide each component of by its magnitude to get the unit vector.

3. Write your answer as .

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Q3. Determine the angle between the vectors and .

Background

Topic: Angle Between Vectors

This question tests your ability to use the dot product to find the angle between two vectors in .

Key Terms and Formula:

• Dot product:

• Angle formula:

1. Compute by multiplying corresponding components and adding.

2. Compute and using the magnitude formula.

3. Plug these values into the angle formula to solve for .

4. Take the arccosine to find (in radians or degrees as required).

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Q4. Compute .

Background

Topic: Dot Product of Vectors

This question asks you to compute the dot product of two vectors written in terms of .

Key Terms and Formula:

• Dot product:

1. Identify the components: , .

2. Multiply corresponding components: , , .

3. Add the results to get the dot product.

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Q5. Find the orthogonal decomposition of as the sum of a vector parallel to and a vector orthogonal to .

Background

Topic: Orthogonal Decomposition of Vectors

This question tests your ability to decompose a vector into components parallel and perpendicular to another vector.

Key Terms and Formula:

• Projection of onto :

• Orthogonal component:

1. Compute and .

2. Find the projection of onto using the formula above.

3. Subtract the projection from to get the orthogonal component.

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Q6. Compute for , .

Background

Topic: Cross Product

This question tests your ability to compute the cross product of two vectors in .

Key Terms and Formula:

• Cross product:

1. Identify the components of and .

2. Apply the cross product formula to compute each component.

3. Write the resulting vector in form.

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