To find the difference [tex]\( u - v \)[/tex] and represent it graphically in two different ways, follow the step-by-step solution:
1. Given Vectors:
- [tex]\( u = \langle 8, 1 \rangle \)[/tex]
- [tex]\( v = \langle 1, -2 \rangle \)[/tex]
2. Find the Difference [tex]\( u - v \)[/tex]:
- To subtract vectors, we subtract corresponding components:
[tex]\[
u - v = \langle u_1 - v_1, u_2 - v_2 \rangle
\][/tex]
- Substituting in the values for [tex]\( u \)[/tex] and [tex]\( v \)[/tex]:
[tex]\[
u - v = \langle 8 - 1, 1 - (-2) \rangle
\][/tex]
[tex]\[
u - v = \langle 7, 3 \rangle
\][/tex]
3. Negate Vector [tex]\( v \)[/tex]:
- To negate a vector, negate each of its components:
[tex]\[
-v = \langle -v_1, -v_2 \rangle
\][/tex]
- Substituting in the values for [tex]\( v \)[/tex]:
[tex]\[
-v = \langle -1, 2 \rangle
\][/tex]
4. Find [tex]\( u + (-v) \)[/tex]:
- To add vectors, we add corresponding components:
[tex]\[
u + (-v) = \langle u_1 + (-v_1), u_2 + (-v_2) \rangle
\][/tex]
- Substituting in the values for [tex]\( u \)[/tex] and [tex]\( -v \)[/tex]:
[tex]\[
u + (-v) = \langle 8 + (-1), 1 + 2 \rangle
\][/tex]
[tex]\[
u + (-v) = \langle 7, 3 \rangle
\][/tex]
Therefore, by either subtracting [tex]\( v \)[/tex] from [tex]\( u \)[/tex] directly or by adding [tex]\( u \)[/tex] to the negation of [tex]\( v \)[/tex], we arrive at the same result:
[tex]\[
u - v = \langle 7, 3 \rangle
\][/tex]
This confirms the calculation that the difference [tex]\( u - v \)[/tex] is [tex]\(\langle 7, 3 \rangle\)[/tex].
Graphical Representation:
1. Direct Subtraction Method:
- Draw both vectors [tex]\( u \)[/tex] and [tex]\( v \)[/tex] originating from the same point.
- Draw the vector from the head of [tex]\( v \)[/tex] to the head of [tex]\( u \)[/tex]. This is the vector [tex]\( u - v \)[/tex].
- This vector should end up being [tex]\(\langle 7, 3 \rangle\)[/tex].
2. Addition of Negated Vector Method:
- First, draw vector [tex]\( u \)[/tex].
- Then, draw [tex]\( -v \)[/tex] from the head (endpoint) of [tex]\( u \)[/tex].
- The resulting vector from the tail of [tex]\( u \)[/tex] to the head of [tex]\( -v \)[/tex] is the same as [tex]\( u - v \)[/tex].
- This should also be the vector [tex]\(\langle 7, 3 \rangle\)[/tex].
In summary, [tex]\( u - v \)[/tex] is [tex]\(\langle 7, 3 \rangle\)[/tex] which can be visualized by either subtracting [tex]\( v \)[/tex] from [tex]\( u \)[/tex] directly or by adding [tex]\( u \)[/tex] to the negation of [tex]\( v \)[/tex], and both graphical methods should produce the same resulting vector.
