Understanding Vector Perpendicularity and the Cosine Formula

When dealing with vectors, one fundamental concept is perpendicularity. Two vectors are considered perpendicular (or orthogonal) if they are at a right angle to each other. This means the angle (theta) between them is 90 degrees.

The relationship between vectors and their perpendicularity can be explored using the cosine formula, which is a powerful tool in vector analysis. This article delves into the details of calculating the angle between two vectors using the cosine method and understanding what it means when the angle is 90 degrees.

The Cosine Formula for Vector Dot Product

The dot product of two vectors can be used to determine the angle between them. The dot product formula is defined as:

For two vectors (mathbf{A} a_1mathbf{i} b_1mathbf{j}) and (mathbf{B} a_2mathbf{i} b_2mathbf{j}), the dot product (mathbf{A} cdot mathbf{B}) can be calculated as:

[mathbf{A} cdot mathbf{B} a_1a_2 b_1b_2]

Calculating the Angle Between Vectors

The angle between two vectors can be found using the cosine formula. The formula relating the dot product and the magnitudes of the vectors to the cosine of the angle between them is:

[cos theta frac{mathbf{A} cdot mathbf{B}}{|mathbf{A}| |mathbf{B}|}]

Where (|mathbf{A}|) and (|mathbf{B}|) are the magnitudes (or lengths) of vectors (mathbf{A}) and (mathbf{B}), respectively. The magnitudes are calculated as:

[|mathbf{A}| sqrt{a_1^2 b_1^2}]

[|mathbf{B}| sqrt{a_2^2 b_2^2}]

Example: Calculating the Angle Between Two Vectors

Let's consider two vectors (mathbf{A} 3mathbf{i} 4mathbf{j}) and (mathbf{B} 5mathbf{i} - 4mathbf{j}).

First, we calculate the dot product:

[mathbf{A} cdot mathbf{B} (3)(5) (4)(-4) 15 - 16 -1]

Next, we find the magnitudes of (mathbf{A}) and (mathbf{B}):

[|mathbf{A}| sqrt{3^2 4^2} sqrt{9 16} sqrt{25} 5]

[|mathbf{B}| sqrt{5^2 (-4)^2} sqrt{25 16} sqrt{41}]

Using the cosine formula:

[cos theta frac{-1}{5 sqrt{41}}]

This means the angle (theta) can be found using the inverse cosine function:

[theta cos^{-1} left( frac{-1}{5sqrt{41}} right)]

When the Angle is 90 Degrees

An interesting case arises when the angle between two vectors is 90 degrees. In this scenario, the vectors are perpendicular to each other.

The cosine formula reveals that when the angle (theta 90^circ), the cosine of 90 degrees is 0. This implies:

[cos 90^circ 0]

Which means:

[frac{mathbf{A} cdot mathbf{B}}{|mathbf{A}| |mathbf{B}|} 0]

This will hold true if the dot product of the vectors is 0, i.e., (mathbf{A} cdot mathbf{B} 0). This condition is sufficient to assert that the vectors are perpendicular.

Conclusion

The cosine formula is a crucial tool in vector analysis, helping us understand the relationship between vectors and the angles they form. Understanding these concepts is essential for various applications in physics, engineering, and mathematics.

Perpendicular vectors, indicated by the cosine of 90 degrees being 0, play a vital role in these fields. They are used in calculating projections, determining orthogonality, and solving complex problems involving vector spaces.