Use vectors to decide whether the triangle with vertices
P(3, −3, 0), Q(4, 0, −2), and R(8, −2, −3)
is right-angled.
I feel like this uses the cross product but I'm not sure how.
PQ^2=1+9+4=14
PR^2=25+1+9=35
QR^2=16+4+1=21
To determine whether a triangle is right-angled using vectors, you can check if the dot product of any two sides is zero. If the dot product is zero, it means the two sides are orthogonal, indicating that the triangle is right-angled.
Here's how you can do it step-by-step:
1. Find the position vectors of each point:
- Position vector of P: p = (3, -3, 0)
- Position vector of Q: q = (4, 0, -2)
- Position vector of R: r = (8, -2, -3)
2. Calculate the direction vectors of two sides of the triangle:
- Direction vector of PQ: v1 = q - p = (4 - 3, 0 - (-3), -2 - 0) = (1, 3, -2)
- Direction vector of PR: v2 = r - p = (8 - 3, -2 - (-3), -3 - 0) = (5, 1, -3)
3. Compute the dot product between v1 and v2:
- v1 · v2 = (1 * 5) + (3 * 1) + (-2 * -3)
= 5 + 3 + 6
= 14
4. Check if the dot product is zero:
- Since v1 · v2 = 14 ≠ 0, the triangle is not right-angled.
In this case, the dot product of the two direction vectors is not zero, so the triangle formed by points P, Q, and R is not right-angled.