For the following vectors:

[tex]\[ v = 7i + 4j \][/tex]
[tex]\[ w = 4i - 7j \][/tex]

(a) Find the dot product [tex]\( v \cdot w \)[/tex].

(b) Find the angle between [tex]\( v \)[/tex] and [tex]\( w \)[/tex].

(c) State whether the vectors are parallel, orthogonal, or neither.

(a) [tex]\( v \cdot w = 0 \)[/tex] (Simplify your answer.)

(b) The angle between [tex]\( v \)[/tex] and [tex]\( w \)[/tex] is [tex]\( \theta = \square^{\circ} \)[/tex]. (Simplify your answer.)



Answer :

Absolutely, let's start solving the given problem step-by-step.

Given vectors [tex]\( v = 7\mathbf{i} + 4\mathbf{j} \)[/tex] and [tex]\( w = 4\mathbf{i} - 7\mathbf{j} \)[/tex]:

### (a) Finding the Dot Product [tex]\( v \cdot w \)[/tex]

The dot product of two vectors [tex]\( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} \)[/tex] and [tex]\( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} \)[/tex] is calculated using the formula:
[tex]\[ v \cdot w = a_1b_1 + a_2b_2 \][/tex]

For our vectors:
[tex]\[ v \cdot w = (7)(4) + (4)(-7) \][/tex]
[tex]\[ v \cdot w = 28 - 28 \][/tex]
[tex]\[ v \cdot w = 0 \][/tex]

Thus, the dot product [tex]\( v \cdot w = 0 \)[/tex].

### (b) Finding the Angle Between [tex]\( v \)[/tex] and [tex]\( w \)[/tex]

The angle [tex]\( \theta \)[/tex] between two vectors can be found using the formula:
[tex]\[ \cos(\theta) = \frac{v \cdot w}{\|v\| \|w\|} \][/tex]

However, if the dot product of two vectors is zero, the vectors are orthogonal (perpendicular) and the angle between them is 90 degrees.

Since we found [tex]\( v \cdot w = 0 \)[/tex], the angle [tex]\( \theta \)[/tex] between [tex]\( v \)[/tex] and [tex]\( w \)[/tex] is:
[tex]\[ \theta = 90^\circ \][/tex]

### (c) Determining Whether the Vectors Are Parallel, Orthogonal, or Neither

To determine the relationship between the vectors, we analyze the dot product result:

- Parallel Vectors: Vectors are parallel if one is a scalar multiple of the other.
- Orthogonal Vectors: Vectors are orthogonal if their dot product is zero.
- Neither: If they aren't parallel or orthogonal.

From our calculations:
- The dot product [tex]\( v \cdot w = 0 \)[/tex]

Thus, the vectors are orthogonal.

### Summary:
(a) The dot product [tex]\( v \cdot w \)[/tex] is [tex]\( 0 \)[/tex].

(b) The angle between [tex]\( v \)[/tex] and [tex]\( w \)[/tex] is [tex]\( \theta = 90^\circ \)[/tex].

(c) The vectors are orthogonal.