Given that P=(5,4), Q=(7,3), R=(8,6), and S=(4,1). Find the component form of the vector PQ+5RS
How would I work that out?
To find the component form of the vector PQ+5RS, we need to calculate the vector PQ and the vector RS, and then add them together.
1. Calculate the vector PQ:
- PQ = (x2 - x1, y2 - y1)
- For PQ, given P=(5,4) and Q=(7,3):
- x2 - x1: 7 - 5 = 2
- y2 - y1: 3 - 4 = -1
- Therefore, PQ = (2, -1)
2. Calculate the vector RS:
- RS = (x2 - x1, y2 - y1)
- For RS, given R=(8,6) and S=(4,1):
- x2 - x1: 4 - 8 = -4
- y2 - y1: 1 - 6 = -5
- Therefore, RS = (-4, -5)
3. Multiply RS by 5:
- 5RS = (5 * -4, 5 * -5)
- 5RS = (-20, -25)
4. Add the vectors PQ and 5RS:
- PQ + 5RS = (PQx + 5RSx, PQy + 5RSy)
- For PQ + 5RS, substituting the values we calculated:
- PQx + 5RSx: 2 + (-20) = -18
- PQy + 5RSy: -1 + (-25) = -26
- Therefore, PQ + 5RS = (-18, -26)
Hence, the component form of the vector PQ+5RS is (-18, -26).