Angle Between Vectors a and b: A Comprehensive Guide

Understanding the angle between two vectors is an essential concept in vector mathematics. This article will delve into the detailed calculation of the angle between the vectors ( a 3i - 2j 10k ) and ( b 7i 4j 4k ) using the dot product method.

Introduction to Vectors and Dot Product

Vectors are fundamental in mathematics and physics, representing entities that have both magnitude and direction. The dot product of two vectors, denoted as ( a cdot b ), is a scalar quantity that can be used to determine the angle between them. The formula for the dot product is given by:

Dot Product Formula

The dot product of vectors ( a ) and ( b ) is calculated as follows:

[latex]mathbf{a} cdot mathbf{b} a_x b_x a_y b_y a_z b_z[/latex]

Where ( a_x, a_y, a_z ) and ( b_x, b_y, b_z ) are the components of vectors ( a ) and ( b ). The dot product can also be expressed in terms of the magnitudes of the vectors and the cosine of the angle ( theta ) between them:

[latex]mathbf{a} cdot mathbf{b} ||mathbf{a}|| ||mathbf{b}|| cos(theta)[/latex]

Step-by-Step Calculation of the Angle

Let's break down the calculation of the angle between the vectors ( a 3i - 2j 10k ) and ( b 7i 4j 4k ).

Step 1: Calculate the Dot Product (a.b)

Using the dot product formula:

[latex]mathbf{a} cdot mathbf{b} (3)(7) (-2)(4) (10)(4) 21 - 8 40 53[/latex]

Step 2: Calculate the Magnitudes of Vectors a and b

The magnitude of vector ( a ) is:

[latex]||mathbf{a}|| sqrt{3^2 (-2)^2 10^2} sqrt{9 4 100} sqrt{113}[/latex]

The magnitude of vector ( b ) is:

[latex]||mathbf{b}|| sqrt{7^2 4^2 4^2} sqrt{49 16 16} sqrt{81} 9[/latex]

Step 3: Use the Dot Product and Magnitudes to Find Theta

Now, using the dot product and magnitudes, we can find the angle ( theta ): [latex]mathbf{a} cdot mathbf{b} ||mathbf{a}|| ||mathbf{b}|| cos(theta) Rightarrow 53 sqrt{113} cdot 9 cos(theta) Rightarrow cos(theta) frac{53}{9sqrt{113}} Rightarrow theta cos^{-1} left( frac{53}{9sqrt{113}} right)[/latex]

Conclusion and Applications

The angle ( theta ) between the vectors ( a 3i - 2j 10k ) and ( b 7i 4j 4k ) is given by ( theta cos^{-1} left( frac{53}{9sqrt{113}} right) ). The angle calculation is a crucial component in various fields, such as physics, engineering, and computer graphics, where vector quantities are used to describe and analyze physical phenomena and geometric configurations.

Key Takeaways

- The dot product of vectors can be used to determine the angle between them. - The formula for the dot product is ( a cdot b a_x b_x a_y b_y a_z b_z ). - The cosine of the angle between two vectors can be calculated using the magnitudes of the vectors and their dot product.

Related Keywords

angle between vectors, vector dot product, vector multiplication