Find the vectors and whose initial and terminal points are given

Question

,
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!