Question

Let P(3,2,−1),Q(−2,1,c) and R(c,1,0) be points in R 3 . (a) Use the cross product definition to find the area of triangle △PQR in terms of c. (b) For what values of c (if any) is △PQR a right triangle?

Answer 1

(a) The area of triangle △PQR in terms of
`\(c\)` is given by the magnitude of the cross product
`\(\overrightarrow{PQ} * \overrightarrow{PR}\)`. The area is
`\(|c + 1|√(c^2 + 2c + 6)\).`

(b) △PQR is a right triangle when the dot product
`\(\overrightarrow{PQ} \cdot \overrightarrow{PR}\)` is zero. Solving this, we find
`\(c = -3\) or \(c = -1\).`

Step-by-step explanation:

(a) The cross product
`\(\overrightarrow{PQ} * \overrightarrow{PR}\)`can be calculated using the determinant of the matrix formed by the unit vectors
`\(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\)`, and the direction vectors of
`\(\overrightarrow{PQ}\)` and
`\(\overrightarrow{PR}\).` The magnitude of the cross product is then computed, yielding the area of the triangle in terms of
`\(c\).`

(b) For △PQR to be a right triangle, the dot product
`\(\overrightarrow{PQ} \cdot \overrightarrow{PR}\)` must be zero. Setting up and solving this equation, we find the values of
`\(c\)` for which the triangle is a right triangle. In this case,
`\(c = -3\) or \(c = -1\).`

In conclusion, the area of △PQR is expressed in terms of
`\(c\)`, and △PQR is a right triangle when
`\(c = -3\) or \(c = -1\).` These findings provide geometric insights into the properties of the triangle formed by the given points in three-dimensional space.