Question

Find the vectors and whose initial and terminal points are given
,
Determine whether the located vectors and are parallel.
(a) , , ,
(b) , , ,
Find and if .
Let , and . Find
b. c. d.
e. f. the angle between and g.
h. i. the unit vector in the direction of
a. Determine such that where .
If and are perpendicular unit vectors, show that .
a. Given that , . Determine the value of for which the vectors and will be mutually perpendicular.
Find a real number such that the vectors and are perpendicular.
Vectors and make an angle of . If and then calculate the angle between the vectors and .
a. Find the projection of vector along vector .
b. Find the cosine of the angle between the vectors and .
c. Find the direction cosines of the vector .
Let , and be vectors in . Then: .
Each of the angles between the vectors is . If , and , find the norm of the vector .
Given three vectors such that if , and . Evaluate .
Prove that if and are vectors in , then
(a) (b)
(c) if and only if and have the same direction.
Given three vectors , and be three non-zero vectors. If , is it

Answer 1

To help clarify your request, I will break down the parts of your question based on common topics in vector mathematics. I will also provide answers or guidance where appropriate.

1. **Finding Vectors**:
If you're given initial points \( A(x_1, y_1, z_1) \) and terminal points \( B(x_2, y_2, z_2) \), the vector \( \vec{AB} \) can be expressed as:
\[
\vec{AB} = (x_2 - x_1, y_2 - y_1, z_2 - z_1)
\]

2. **Determining Parallelism of Vectors**:
Two vectors \( \vec{a} \) and \( \vec{b} \) are parallel if there exists a scalar \( k \) such that \( \vec{b} = k \vec{a} \). Alternatively, they can be parallel if:
\[
\frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3} \quad (\text{assuming none are zero})
\]

3. **Creating and Using Normalized Vectors**:
The unit vector in the direction of \( \vec{v} \) is given by:
\[
\hat{v} = \frac{\vec{v}}{\|\vec{v}\|}
\]
where \( \|\vec{v}\| = \sqrt{x^2 + y^2 + z^2} \).

4. **Angle Between Two Vectors**:
The cosine of the angle \( \theta \) between two vectors \( \vec{a} \) and \( \vec{b} \) is found using the dot product:
\[
\cos \theta = \frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\| \|\vec{b}\|}
\]

5. **Projection of One Vector onto Another**:
The projection of vector \( \vec{a} \) onto vector \( \vec{b} \) is given by:
\[
\text{proj}_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{\|\vec{b}\|^2} \vec{b}
\]

6. **Direction Cosines**:
For a vector \( \vec{v} = (x, y, z) \), the direction cosines are:
\[
\cos \alpha = \frac{x}{\|\vec{v}\|}, \quad \cos \beta = \frac{y}{\|\vec{v}\|}, \quad \cos \gamma = \frac{z}{\|\vec{v}\|}
\]

7. **Mutually Perpendicular Vectors**:
Two vectors \( \vec{a} \) and \( \vec{b} \) are perpendicular if their dot product is zero:
\[
\vec{a} \cdot \vec{b} = 0
\]

8. **Checking if \( \vec{v} = c\vec{w} \) indicates parallel directions**:
If \( \vec{v} \) and \( \vec{w} \) are non-zero vectors and \( \vec{v} = c\vec{w} \), then both vectors are in the same direction (or opposite direction if \( c < 0 \)).

Given the various parts of your query, would you like to focus on a specific subproblem (such as finding angles, projections, or parallelism), or provide numerical examples to clarify these concepts? Let me know how you would like to proceed!