Question

Given that \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}=\overrightarrow{\mathbf{C}}\) and that \(A^{2}+B^{2}=C^{2}\), how are \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\) oriented relative to one another?

Short answer

\(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\) are perpendicular.

Step-by-step solution

Understand the Given Information

We are given two equations: \(\overrightarrow{\mathbf{A}} + \overrightarrow{\mathbf{B}} = \overrightarrow{\mathbf{C}}\) and \(A^2 + B^2 = C^2\). Here, \(A\), \(B\), and \(C\) represent the magnitudes of the vectors \(\overrightarrow{\mathbf{A}}\), \(\overrightarrow{\mathbf{B}}\), and \(\overrightarrow{\mathbf{C}}\), respectively.

Recall the Triangle Relationship

The vectors \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\), when added, form the vector \(\overrightarrow{\mathbf{C}}\). This suggests a relationship involving a triangle, where \(\overrightarrow{\mathbf{A}}\), \(\overrightarrow{\mathbf{B}}\), and \(\overrightarrow{\mathbf{C}}\) follow the triangle inequality rule.

Apply Pythagorean Theorem

The equation \(A^2 + B^2 = C^2\) suggests the applicability of the Pythagorean theorem. This indicates that \(\overrightarrow{\mathbf{A}}\), \(\overrightarrow{\mathbf{B}}\), and \(\overrightarrow{\mathbf{C}}\) must form a right triangle.

Conclude Vector Orientation

Since \(A^2 + B^2 = C^2\) follows the Pythagorean theorem, it confirms that \(\overrightarrow{\mathbf{A}}\) is perpendicular to \(\overrightarrow{\mathbf{B}}\). Therefore, \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\) are oriented at a right angle (90 degrees) to one another.

Key concepts

Right Angle

In geometry, a right angle is an angle of exactly 90 degrees. It is formed when two lines or vectors intersect perpendicularly, creating an "L" shape. Understanding right angles is essential in vector mathematics, especially in problems dealing with vector addition and orientation.
When vectors are said to be at a right angle to each other, it means that they intersect at a point, creating an angle of 90 degrees between them. This perpendicular relationship is quite significant because it impacts how vectors interact and combine.

• Two vectors positioned at a right angle with respect to each other have no impact on one another’s magnitude in their respective directions.
• This orientation is common in many practical applications, such as physics problems involving components of forces.

In the context of vector addition and the given exercise, establishing the presence of a right angle helps confirm the relationship between the vectors involved.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in mathematics, especially relevant when dealing with right triangles. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
This can be expressed as:\[c^2 = a^2 + b^2\]
where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides.
In vector terms, when vectors form a right triangle, the same principle applies, where the resultant vector (often called \(\overrightarrow{\mathbf{C}}\)) serves as the hypotenuse.

• The equation \(A^2 + B^2 = C^2\) from the exercise is a direct result of the Pythagorean Theorem, indicating the presence of a right triangle.
• It confirms that vectors \(\overrightarrow{\mathbf{A}}\) and \(\overrightarrow{\mathbf{B}}\) form legs of the triangle, and \(\overrightarrow{\mathbf{C}}\) is the hypotenuse.

This theorem not only establishes the angular relationships between the vectors but also provides a method to compute the resultant vector's magnitude.

Vector Magnitude

The magnitude of a vector is a measure of its length or size. It is an essential concept when calculating and understanding the effects of vector quantities.
Magnitude can be calculated using Pythagorean Theorem principles when vectors are oriented at a right angle, as introduced in the problem.

• For a vector \(\overrightarrow{\mathbf{V}} = (x, y)\), its magnitude can be determined as \(\sqrt{x^2 + y^2}\).
• In three-dimensional space or higher, similar methods apply using additional vector components.

In the context of the exercise, the magnitudes of \(\overrightarrow{\mathbf{A}}\), \(\overrightarrow{\mathbf{B}}\), and \(\overrightarrow{\mathbf{C}}\) fit the Pythagorean relationship of right triangles, helping to confirm their 90-degree relationship.
Understanding vector magnitudes is critical for correctly interpreting vector addition and orientation, as it ensures that calculations align with the spatial configuration of vectors involved in complex problems.