A triangle has vertices A(2,3,7), B(0,-3,4) and C(5,2,-4)
a.) Determine the largest angle in the triangle.
To determine the largest angle in a triangle, we need to find the three angles of the triangle and compare them. The largest angle will be the one with the largest measure.
To find the angles of the triangle, we can use the concept of dot products. The dot product of two vectors can be used to find the angle between them.
Let's start by finding the vectors from A to B and A to C:
Vector AB: B - A
= (0, -3, 4) - (2, 3, 7)
= (-2, -6, -3)
Vector AC: C - A
= (5, 2, -4) - (2, 3, 7)
= (3, -1, -11)
Next, we can calculate the dot product of these vectors. The dot product of two vectors, say u and v, is calculated as: u · v = u1 * v1 + u2 * v2 + u3 * v3.
Dot product of AB and AC:
(AB) · (AC) = (-2 * 3) + (-6 * -1) + (-3 * -11)
= -6 + 6 + 33
= 33
To find the magnitude (length) of a vector, we use the formula ||v|| = √(v1^2 + v2^2 + v3^2).
Magnitude of AB:
||AB|| = √((-2)^2 + (-6)^2 + (-3)^2)
= √(4 + 36 + 9)
= √49
= 7
Magnitude of AC:
||AC|| = √(3^2 + (-1)^2 + (-11)^2)
= √(9 + 1 + 121)
= √131
Now, we can calculate the angle using the formula:
cosθ = (AB · AC) / (||AB|| * ||AC||)
Let's plug in the values we have calculated:
cosθ = 33 / (7 * √131)
Using a calculator, we find cosθ ≈ 0.717
To find the angle θ, we can take the inverse cosine (cos^(-1)) of the value. In this case:
θ ≈ cos^(-1)(0.717)
Using a calculator, we find θ ≈ 45.57 degrees.
Similarly, we can find the angles formed at the other two vertices, B and C. Let's call them angle B and angle C.
To find angle B, we can use the vectors BA and BC and repeat the above process.
To find angle C, we can use the vectors CA and CB and follow the same procedure.
Once we have all three angles, we can compare them to determine the largest angle in the triangle.