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.
