Question

Show that the triangle with vertices \(A(4,-7,9), B(6,4,4),\) and \(C(7,10,-6)\) is not a rightangled triangle.

Short answer

The triangle is not right-angled as none of the dot products is zero.

Step-by-step solution

Determine the vectors representing the triangle sides

First, we need to find the vectors that form the sides of the triangle. We calculate these by subtracting the coordinates of the points. For vector \(\overrightarrow{AB}\), subtract the coordinates of \(A\) from \(B\):\[\overrightarrow{AB} = B - A = (6 - 4, 4 + 7, 4 - 9) = (2, 11, -5)\]. Similarly, for \(\overrightarrow{BC}\):\[\overrightarrow{BC} = C - B = (7 - 6, 10 - 4, -6 - 4) = (1, 6, -10)\] and for \(\overrightarrow{AC}\):\[\overrightarrow{AC} = C - A = (7 - 4, 10 + 7, -6 - 9) = (3, 17, -15)\].

Calculate the dot products

A triangle is right-angled if one of the dot products of its vectors is zero, indicating perpendicularity. Calculate the dot products:\[\overrightarrow{AB} \cdot \overrightarrow{BC} = (2)(1) + (11)(6) + (-5)(-10) = 2 + 66 + 50 = 118\], \[\overrightarrow{AB} \cdot \overrightarrow{AC} = (2)(3) + (11)(17) + (-5)(-15) = 6 + 187 + 75 = 268\], \[\overrightarrow{BC} \cdot \overrightarrow{AC} = (1)(3) + (6)(17) + (-10)(-15) = 3 + 102 + 150 = 255\].

Analyze the results

None of the dot products \(\overrightarrow{AB} \cdot \overrightarrow{BC} = 118\), \(\overrightarrow{AB} \cdot \overrightarrow{AC} = 268\), or \(\overrightarrow{BC} \cdot \overrightarrow{AC} = 255\) is zero. Therefore, none of the angles formed between the vectors is 90°.

Key concepts

Vectors

Vectors are essential in understanding geometry. They represent quantities that have both magnitude and direction. In the context of a triangle, vectors can represent the sides or edges of the triangle. For instance, if we have a triangle with vertices at points \(A\), \(B\), and \(C\), the vector \(\overrightarrow{AB}\) can be obtained by subtracting the coordinates of \(A\) from \(B\). This results in a new vector that tells us how we move from point \(A\) to point \(B\). Such calculations for \(\overrightarrow{AB}, \overrightarrow{BC},\) and \(\overrightarrow{AC}\) help in visualizing and analyzing the spatial relationships between the points. Each vector is represented by an ordered triplet of numbers, which correspond to its projection on the coordinate axes. This makes computation straightforward and enables further analysis using mathematical operations.

Dot Product

The dot product is a crucial mathematical operation used to determine the relationship between two vectors. It gives us a scalar quantity when two vectors are multiplied together. Mathematically, for vectors \(\mathbf{u} = (u_1, u_2, u_3)\) and \(\mathbf{v} = (v_1, v_2, v_3)\), the dot product \(\mathbf{u} \cdot \mathbf{v}\) is calculated as \(u_1 v_1 + u_2 v_2 + u_3 v_3\).
This operation is particularly useful in determining the angle between two vectors. When the dot product is zero, the vectors are perpendicular to each other. This indicates a 90-degree angle between them, which is a characteristic feature of right angles. In this exercise, the dot products between vectors \(\overrightarrow{AB}\), \(\overrightarrow{BC}\), and \(\overrightarrow{AC}\) are calculated to check if any two vectors are perpendicular, hence forming a right angle in the triangle.

Right-Angle Triangle

A right-angle triangle is a triangle in which one of the angles is exactly 90 degrees. This kind of triangle has unique properties, especially concerning its sides and angles. To determine if a triangle is a right-angle triangle using vectors, we check if any of the dot products of its side vectors is zero. This condition signifies that the vectors are orthogonal, forming a right angle between them. However, if no pair of vectors in the triangle results in a dot product of zero, the triangle is not right-angled.
The exercise demonstrated this concept by calculating the dot products of \(\overrightarrow{AB} \cdot \overrightarrow{BC}\), \(\overrightarrow{AB} \cdot \overrightarrow{AC}\), and \(\overrightarrow{BC} \cdot \overrightarrow{AC}\). Since none of these dot products equaled zero, it concluded that the given triangle is not a right-angle triangle.