Problem 18 - Calculus: Early Transcendentals

Question

Do the three points \((1,2,0),(-2,1,1),\) and (0,3,-1) form a right triangle?

Step-by-Step Solution

Answer

The three points do not form a right triangle as no vectors are perpendicular.

Step 1: Calculate Vectors

To determine if the given points form a right triangle, we will first calculate the vectors between the points. Let's consider the vectors: \( \mathbf{AB} \) from \((1,2,0)\) to \((-2,1,1)\), vector \( \mathbf{BC} \) from \((-2,1,1)\) to \((0,3,-1)\), and vector \( \mathbf{CA} \) from \((0,3,-1)\) to \((1,2,0)\).\[\mathbf{AB} = (-2-1, 1-2, 1-0) = (-3, -1, 1)\]\[\mathbf{BC} = (0-(-2), 3-1, -1-1) = (2, 2, -2)\]\[\mathbf{CA} = (1-0, 2-3, 0-(-1)) = (1, -1, 1)\]

Step 2: Determine Dot Products

The dot product of two vectors will be zero if and only if the vectors are perpendicular (form a 90-degree angle). Calculate the dot products: \( \mathbf{AB} \cdot \mathbf{BC} \), \( \mathbf{BC} \cdot \mathbf{CA} \), and \( \mathbf{CA} \cdot \mathbf{AB} \).\[\mathbf{AB} \cdot \mathbf{BC} = (-3)(2) + (-1)(2) + (1)(-2) = -6 - 2 - 2 = -10\]\[\mathbf{BC} \cdot \mathbf{CA} = (2)(1) + (2)(-1) + (-2)(1) = 2 - 2 - 2 = -2\]\[\mathbf{CA} \cdot \mathbf{AB} = (1)(-3) + (-1)(-1) + (1)(1) = -3 + 1 + 1 = -1\]

Step 3: Conclusion Based on Dot Products

Neither of the calculated dot products is zero, meaning none of the vectors is perpendicular to another. Therefore, the points do not form a right triangle because there is no right angle present.

Key Concepts

Vectors

Vectors are incredibly useful in mathematics, especially in geometry and physics. They help us represent quantities that have both magnitude and direction, like force or velocity. A vector can be thought of as an arrow connecting two points. In a right triangle problem, we can use vectors to represent the sides of the triangle formed by the given points. To see if these points form a right triangle, we need to calculate vectors between each pair of points. The vectors tell us the direction and distance between them, and later we will use these vector calculations to check the angles between sides.

Dot Product

The dot product, also known as the scalar product, is a mathematical operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This product describes the magnitude of one vector projected onto another, which is crucial for determining angles between vectors.

The formula for the dot product of two vectors \(\mathbf{A} = (a_1, a_2, a_3)\) and \(\mathbf{B} = (b_1, b_2, b_3)\) is given by:

• \(\mathbf{A} \cdot \mathbf{B} = a_1 b_1 + a_2 b_2 + a_3 b_3\)

The result will determine if the vectors are perpendicular — that is, they form a right angle. For this, the dot product must equal zero. If it's not zero, like in our problem above, the vectors are not perpendicular.

Perpendicular Vectors

Perpendicular vectors are vectors that intersect at a 90-degree angle. This concept is key in geometry because perpendicular lines or vectors help form right angles, crucial for identifying right triangles. In vector terms, two vectors being perpendicular means their dot product equals zero.

This zero result occurs because when two vectors are at right angles to each other, the projection of one vector onto the other is zero, reflecting no overlap along any axis. If any vectors in a triangle are perpendicular, it would mean that the triangle formed is a right triangle. But, if the dot product result is not zero, like in this case, it confirms the absence of a right angle, indicating that the triangle is not a right triangle.