Question

Find the angle \(A B C\) if the points are \(A(1,2,3)\), \(B(-4,5,6)\), and \(C(1,0,1)\)

Short answer

The angle \( \angle ABC \) is approximately \( 138.59^\circ \).

Step-by-step solution

Find Vector AB

First, we find the vector \( \overrightarrow{AB} \), which is obtained by subtracting coordinates of \( A \) from coordinates of \( B \). \[ \overrightarrow{AB} = B - A = (-4 - 1, 5 - 2, 6 - 3) = (-5, 3, 3) \]

Find Vector BC

Next, we find the vector \( \overrightarrow{BC} \), by subtracting the coordinates of \( B \) from the coordinates of \( C \).\[ \overrightarrow{BC} = C - B = (1 + 4, 0 - 5, 1 - 6) = (5, -5, -5) \]

Calculate Dot Product of AB and BC

The dot product of two vectors \( \overrightarrow{AB} \) and \( \overrightarrow{BC} \) is calculated as follows:\[ \overrightarrow{AB} \cdot \overrightarrow{BC} = (-5)(5) + (3)(-5) + (3)(-5) = -25 - 15 - 15 = -55 \]

Calculate Magnitude of Vectors AB and BC

Now, find the magnitude of both vectors. The magnitude of \( \overrightarrow{AB} \):\[ |\overrightarrow{AB}| = \sqrt{(-5)^2 + 3^2 + 3^2} = \sqrt{25 + 9 + 9} = \sqrt{43} \]The magnitude of \( \overrightarrow{BC} \):\[ |\overrightarrow{BC}| = \sqrt{5^2 + (-5)^2 + (-5)^2} = \sqrt{25 + 25 + 25} = \sqrt{75} = 5\sqrt{3} \]

Use Cosine Formula to Find Angle

Using the dot product and magnitudes, calculate the cosine of the angle \( \angle ABC \).\[ \cos(\angle ABC) = \frac{\overrightarrow{AB} \cdot \overrightarrow{BC}}{|\overrightarrow{AB}| |\overrightarrow{BC}|} = \frac{-55}{\sqrt{43} \times 5\sqrt{3}} = \frac{-55}{5\sqrt{129}} \]Therefore, the angle \( \angle ABC \) is:\[ \angle ABC = \cos^{-1}\left(\frac{-55}{5\sqrt{129}}\right) \]

Calculate the Angle

Using a calculator, find \( \angle ABC \).\( \angle ABC \approx \cos^{-1}\left(\frac{-55}{5\sqrt{129}}\right) \). Evaluating this gives:\( \angle ABC \approx 138.59^\circ \)

Key concepts

Dot Product

The dot product, also known as the scalar product, is an operation that takes two vectors and returns a single number. It is a key concept for calculating angles in vector geometry, especially in 3D space.
The dot product is calculated by multiplying corresponding components of two vectors and then adding these products together.

• For vectors \( \mathbf{a} = (a_1, a_2, a_3) \) and \( \mathbf{b} = (b_1, b_2, b_3) \), the dot product \( \mathbf{a} \cdot \mathbf{b} \) is given by:

\[\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3\]
This result is a measure of how much one vector extends in the direction of another. It also tells us if the vectors are orthogonal (dot product zero), pointing in similar directions (positive), or opposite directions (negative).

Magnitude of a Vector

The magnitude of a vector, also known as the length or norm, provides a measure of how long the vector is. It is crucial for normalizing vectors and will help us understand the size of the vectors involved in calculations.
To find the magnitude of a vector \( \mathbf{a} = (a_1, a_2, a_3) \), use the formula:

• \[|\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}\]

The magnitude is always a non-negative number and represents the vector's distance from the origin in space.
In our example, we calculated the magnitudes of \( \overrightarrow{AB} \) and \( \overrightarrow{BC} \), which is essential in the cosine formula to define the angle between them.

Cosine Formula

The cosine formula is a bridge between the dot product and the magnitudes of vectors. It is used to find the cosine of the angle between two vectors.
Using the magnitudes and the dot product from earlier, the cosine of the angle \( \theta \) between vectors \( \mathbf{a} \) and \( \mathbf{b} \) is given by:

• \[\cos(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}\]

This formula not only tells us how aligned the vectors are (perfectly aligned vectors have a cosine of 1 or -1) but also allows us to solve for the angle itself using the inverse cosine function.
The cosine formula is the key step for converting from a geometric relationship to an angle value, as seen in the calculation of angle \( \angle ABC \).

3D Geometry

3D geometry involves studying shapes and figures that exist in three dimensions. This makes computations, such as determining angles, more complex compared to 2D geometry.
Understanding how vectors function in 3D space is essential for visualizing problems involving points like A, B, and C, as in our exercise.

• Vectors can be thought of as arrows starting from the origin and pointing towards a specific point \((x, y, z)\).
• Transformations like finding vectors \(\overrightarrow{AB}\) and \(\overrightarrow{BC}\) are simplified by subtracting coordinates, allowing us to find directional components between points.
• The computation of angles shows how two vectors meet each other in space, facilitated by their dot product and magnitudes.

These concepts are vital for applications in physics, engineering, and computer graphics, where the orientation and interaction of objects in 3D space are examined.