Problem 26 - Calculus for Biology and Medicine

Question

Find the angle between \(\mathbf{x}=[1,-3,2]^{\prime}\) and \(\mathbf{y}=[3,1,-4]^{\prime}\).

Step-by-Step Solution

Answer

The angle between the vectors is approximately 114.9°.

Step 1: Dot Product of Vectors

Calculate the dot product of the vectors \(\mathbf{x}\) and \(\mathbf{y}\). The dot product is given by the formula \(\mathbf{x} \cdot \mathbf{y} = x_1y_1 + x_2y_2 + x_3y_3\). Using the given vectors \(\mathbf{x} = [1, -3, 2]\) and \(\mathbf{y} = [3, 1, -4]\), the dot product is:\[1\times3 + (-3)\times1 + 2\times(-4) = 3 - 3 - 8 = -8.\]

Step 2: Magnitude of Vectors

Find the magnitude (length) of each vector. The magnitude of a vector \(\mathbf{x} = [x_1, x_2, x_3]\) is calculated using the formula \(\|\mathbf{x}\| = \sqrt{x_1^2 + x_2^2 + x_3^2}\).For \(\mathbf{x} = [1, -3, 2]\):\[\|\mathbf{x}\| = \sqrt{1^2 + (-3)^2 + 2^2} = \sqrt{1 + 9 + 4} = \sqrt{14}.\]For \(\mathbf{y} = [3, 1, -4]\):\[\|\mathbf{y}\| = \sqrt{3^2 + 1^2 + (-4)^2} = \sqrt{9 + 1 + 16} = \sqrt{26}.\]

Step 3: Cosine of the Angle

The cosine of the angle \(\theta\) between the vectors is found using the formula \(\cos(\theta) = \frac{\mathbf{x} \cdot \mathbf{y}}{\|\mathbf{x}\| \cdot \|\mathbf{y}\|}\).Substitute the dot product and the magnitudes found in previous steps:\[\cos(\theta) = \frac{-8}{\sqrt{14} \cdot \sqrt{26}}\]Calculate the product of the magnitudes:\[\sqrt{14} \cdot \sqrt{26} = \sqrt{364} = 19.08.\]Thus, the cosine of the angle is:\[\cos(\theta) = \frac{-8}{19.08} \approx -0.419.\]

Step 4: Angle Between Vectors

Use the arccos function to find \(\theta\):\[\theta = \cos^{-1}(-0.419).\]Calculate \(\theta\) using a calculator:\[\theta \approx 114.9^\circ.\]

Step 5: Conclusion

The angle between the vectors \(\mathbf{x} = [1, -3, 2]\) and \(\mathbf{y} = [3, 1, -4]\) is approximately \(114.9^\circ\).

Key Concepts

Dot Product of Vectors

The dot product is a fundamental operation when working with vectors, as it helps determine the relationship between them. For two vectors, say \( \mathbf{x} = [x_1, x_2, x_3] \) and \( \mathbf{y} = [y_1, y_2, y_3] \), the dot product is calculated by multiplying corresponding components and summing them up. The formula is:

• \( \mathbf{x} \cdot \mathbf{y} = x_1y_1 + x_2y_2 + x_3y_3 \)

For our specific vectors, \( \mathbf{x} = [1, -3, 2] \) and \( \mathbf{y} = [3, 1, -4] \), we find:

• \( \mathbf{x} \cdot \mathbf{y} = 1\times3 + (-3)\times1 + 2\times(-4) = 3 - 3 - 8 = -8 \)

In simpler terms, the dot product here is \(-8\). This value is crucial for finding angles between vectors later.

Magnitude of a Vector

Magnitude essentially measures the length or size of a vector. It tells us how long the vector is in the space it occupies. The magnitude of a vector \( \mathbf{x} = [x_1, x_2, x_3] \) is computed using the formula:

• \( \|\mathbf{x}\| = \sqrt{x_1^2 + x_2^2 + x_3^2} \)

This is similar to finding the hypotenuse of a right triangle. For \( \mathbf{x} = [1, -3, 2] \), the magnitude is:

• \( \|\mathbf{x}\| = \sqrt{1^2 + (-3)^2 + 2^2} = \sqrt{14} \)

For \( \mathbf{y} = [3, 1, -4] \), it is:

• \( \|\mathbf{y}\| = \sqrt{3^2 + 1^2 + (-4)^2} = \sqrt{26} \)

Remember, the magnitude provides a scalar value, making it easier to understand and visualize vectors regardless of their direction.

Angle Between Vectors

Finding the angle between vectors involves the dot product and magnitudes we just learned about. The formula to find the angle \( \theta \) is derived from the dot product and can be expressed as:

• \( \cos(\theta) = \frac{\mathbf{x} \cdot \mathbf{y}}{\|\mathbf{x}\| \cdot \|\mathbf{y}\|} \)

This formula shows the cosine of the angle in terms of the vectors' magnitudes and their dot product. Using our example, we calculate:

• \( \cos(\theta) = \frac{-8}{\sqrt{14} \times \sqrt{26}} = \frac{-8}{19.08} \approx -0.419 \)

To find the actual angle, we use the inverse cosine (arccos) function:

• \( \theta = \cos^{-1}(-0.419) \)

This gives us approximately \( 114.9^\circ \) as the angle between the vectors. The cosine value determines whether the angle is acute, right, or obtuse:

• A positive cosine indicates an acute angle.
• A cosine of zero indicates a right angle.
• A negative cosine, as in this case, indicates an obtuse angle.

Understanding these properties can provide deeper insights into the geometric relationship of vectors in space.