Certainly! Let's work through the problem step-by-step.
### Part (a): Finding the Dot Product [tex]\( u \cdot v \)[/tex]
Given vectors [tex]\( u = \langle -2, 2 \rangle \)[/tex] and [tex]\( v = \langle 1, -1 \rangle \)[/tex], the dot product [tex]\( u \cdot v \)[/tex] is calculated as follows:
[tex]\[
u \cdot v = u_1 \cdot v_1 + u_2 \cdot v_2
\][/tex]
Substituting the values:
[tex]\[
u \cdot v = (-2) \cdot 1 + 2 \cdot (-1)
\][/tex]
Calculating each term:
[tex]\[
(-2) \cdot 1 = -2
\][/tex]
[tex]\[
2 \cdot (-1) = -2
\][/tex]
Adding them together:
[tex]\[
u \cdot v = -2 + (-2) = -4
\][/tex]
So, the dot product [tex]\( u \cdot v \)[/tex] is [tex]\(-4\)[/tex].
[tex]\[
u \cdot v = -4
\][/tex]
### Part (b): Finding the Angle Between [tex]\( u \)[/tex] and [tex]\( v \)[/tex]
To find the angle [tex]\( \theta \)[/tex] between vectors [tex]\( u \)[/tex] and [tex]\( v \)[/tex], we use the formula involving the dot product and magnitudes:
[tex]\[
\cos(\theta) = \frac{u \cdot v}{\|u\| \|v\|}
\][/tex]
First, we calculate the magnitudes of [tex]\( u \)[/tex] and [tex]\( v \)[/tex]:
For vector [tex]\( u \)[/tex]:
[tex]\[
\|u\| = \sqrt{u_1^2 + u_2^2} = \sqrt{(-2)^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\][/tex]
For vector [tex]\( v \)[/tex]:
[tex]\[
\|v\| = \sqrt{v_1^2 + v_2^2} = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}
\][/tex]
Substitute these magnitudes and the dot product into the cosine formula:
[tex]\[
\cos(\theta) = \frac{-4}{(2\sqrt{2})(\sqrt{2})}
\][/tex]
Simplify the denominator:
[tex]\[
(2\sqrt{2})(\sqrt{2}) = 2 \cdot 2 = 4
\][/tex]
So, we have:
[tex]\[
\cos(\theta) = \frac{-4}{4} = -1
\][/tex]
To find [tex]\( \theta \)[/tex], we take the inverse cosine (arccos) of [tex]\(-1\)[/tex]:
[tex]\[
\theta = \arccos(-1)
\][/tex]
The angle whose cosine is [tex]\(-1\)[/tex] is [tex]\( 180^\circ \)[/tex]:
[tex]\[
\theta = 180^\circ
\][/tex]
So, the angle between [tex]\( u \)[/tex] and [tex]\( v \)[/tex] to the nearest degree is [tex]\( 180^\circ \)[/tex].
[tex]\[
\theta = 180^\circ
\][/tex]
In summary:
(a) The dot product [tex]\( u \cdot v \)[/tex] is [tex]\(-4\)[/tex].
(b) The angle between [tex]\( u \)[/tex] and [tex]\( v \)[/tex] is [tex]\( 180^\circ \)[/tex].
