Question

Use cross products to determine whether the points \(A, B,\) and C are collinear. $$A(-3,-2,1), B(1,4,7), \text { and } C(4,10,14)$$

Short answer

Answer: No, the points A, B, and C are not collinear.

Step-by-step solution

Define the vectors

Let's define the vectors, using points A and B to create vector AB, and points B and C to create vector BC: $$\overrightarrow{AB} = B - A = (1, 4, 7) - (-3, -2, 1) = (4, 6, 6)$$ $$\overrightarrow{BC} = C - B = (4, 10, 14) - (1, 4, 7) = (3, 6, 7)$$

Find the cross product of AB and BC

Now, let's find the cross product of the two vectors, \(\overrightarrow{AB} \times \overrightarrow{BC}\): $$ \overrightarrow{AB} \times \overrightarrow{BC} = \begin{bmatrix} i & j & k \\ 4 & 6 & 6 \\ 3 & 6 & 7 \end{bmatrix}$$

Evaluate the cross product

Evaluate the determinant of the matrix in Step 2: $$ \overrightarrow{AB} \times \overrightarrow{BC} = i(6 \cdot 7 - 6 \cdot 6) - j(4 \cdot 7 - 6 \cdot 3) + k(4 \cdot 6 - 6 \cdot 3)$$ $$\overrightarrow{AB} \times \overrightarrow{BC} = 6i - 6j + 0k = (6,-6,0)$$

Check if the cross product is the zero vector

We check if the result of the cross product is the zero vector \((0, 0, 0)\). In this case: $$\overrightarrow{AB} \times \overrightarrow{BC} = (6, -6, 0)$$ The cross product is not the zero vector, so the points A, B, and C are not collinear.