Question

If \(\vec{A}\) and \(\vec{B}\) are vectors and \(\vec{B}=-\vec{A},\) which of the following is true? a) The magnitude of \(\vec{B}\) is equal to the negative of the magnitude of \(\vec{A}\). b) \(\vec{A}\) and \(\vec{B}\) are perpendicular. c) The direction angle of \(\vec{B}\) is equal to the direction angle of \(\vec{A}\) plus \(180^{\circ}\) d) \(\vec{A}+\vec{B}=2 \vec{A}\).

Short answer

a) \(|\vec{B}| = 2|\vec{A}|\) b) \(\vec{A}\) is perpendicular to \(\vec{B}\) c) The direction angle of \(\vec{B}\) is equal to the direction angle of \(\vec{A}\) plus \(180^{\circ}\) d) \(\vec{A} + \vec{B} = 2\vec{A}\) Answer: c) The direction angle of \(\vec{B}\) is equal to the direction angle of \(\vec{A}\) plus \(180^{\circ}\).

Step-by-step solution

Determine the magnitudes

Since \(\vec{B} = -\vec{A}\), the magnitudes \(|\vec{B}|\) and \(|\vec{A}|\) must satisfy the relation \(|\vec{B}| = |-1||\vec{A}|\). Since the absolute value of \(-1\) is 1, the magnitudes must be equal, i.e., \(|\vec{B}| = |\vec{A}|\). Thus, the statement "a" is not correct.

Check for perpendicularity

To determine if \(\vec{A}\) and \(\vec{B}\) are perpendicular, we can calculate their dot product. Using the relation \(\vec{B} = -\vec{A}\), we get \(\vec{A}\cdot\vec{B} = \vec{A}\cdot(-\vec{A}) = -\vec{A}\cdot\vec{A} = -|\vec{A}|^2\). Now, if they were perpendicular, the dot product would equal \(0\). However, since both vectors have non-zero magnitudes, the dot product is non-zero, and statement "b" is incorrect.

Compare direction angles

Based on the given relation between vectors, \(\vec{B}\) has the same magnitude as \(\vec{A}\) but is pointing in the opposite direction. Therefore, their direction angles must differ by 180°, either adding or subtracting this value. Thus, statement "c" is correct.

Calculate vector sum

To verify whether statement "d" is correct, we first calculate the vector sum of both \(\vec{A}\) and \(\vec{B}\). Since \(\vec{B} = -\vec{A}\), we get \(\vec{A} + \vec{B} = \vec{A} + (-\vec{A}) = \vec{A} - \vec{A} = \overrightarrow{0}\). This result differs from statement "d", which states that their sum should be \(2\vec{A}\), so this statement is incorrect. From the analysis, the correct choice is "c". The direction angle of \(\vec{B}\) is equal to the direction angle of \(\vec{A}\) plus \(180^{\circ}\).

Key concepts

Magnitude of Vectors

The magnitude of a vector is a measure of its length or size. Think of it as how long the vector is when measured with a ruler. When we write a vector, such as \(\vec{A}\), its magnitude is represented as \(|\vec{A}|\).

In the given exercise, we have two vectors, \(\vec{A}\) and \(\vec{B}\). According to the relation \(\vec{B} = -\vec{A}\), \(\vec{B}\) is just \(\vec{A}\) flipped in the opposite direction. However, this change in direction does not affect length. As a result, the magnitudes are the same.

To put it simply:

• Magnitude is always a positive number that tells us how big a vector is.
• Even if the vector changes direction, its magnitude stays the same.
• In this problem, \(|\vec{B}| = |\vec{A}|\), not negative \(|\vec{A}|\).

Understanding magnitudes is crucial because it allows us to compare one vector with another. It's like comparing the lengths of lines or paths.

Dot Product

The dot product is a way to multiply two vectors together. But unlike regular multiplication, the dot product results in a scalar, which is just a fancy word for a number without direction. This product can tell us if two vectors are "working together" or "against each other."

In mathematical terms, the dot product of two vectors \(\vec{A}\) and \(\vec{B}\) is calculated as:

• \(\vec{A}\cdot\vec{B} = |\vec{A}| |\vec{B}| \cos(\theta)\)

where \(\theta\) is the angle between the two vectors.

In our case, with \(\vec{B} = -\vec{A}\), we substitute to get \(\vec{A}\cdot(-\vec{A}) = -(|\vec{A}|^2)\). The dot product measures how much two vectors point in the same direction. For perpendicular vectors, their dot product is zero. Thus, since \(\vec{A}\cdot\vec{B}\) results in a negative, non-zero value, \(\vec{A}\) and \(\vec{B}\) are not perpendicular.

To summarize:

• Dot product results in a simple number, indicating the degree of alignment.
• If zero, vectors are perpendicular; if positive, they point in the same direction; if negative, they point in opposite directions.
• Here, \(\vec{A}\cdot\vec{B}\) isn't zero, so \(\vec{A}\) and \(\vec{B}\) don't form a right angle.

This concept helps us understand vector properties in physics and engineering.

Direction Angles

When we talk about vectors, direction is just as important as magnitude. The direction angle of a vector is a measure of which way it's pointing. Imagine a compass where north is the initial direction.

In this problem, the vector \(\vec{B}\) is simply \(\vec{A}\) pointed in the opposite direction. So, naturally, this results in the direction angle of \(\vec{B}\) being \(180^{\circ}\) different from \(\vec{A}\). This means while the magnitude (or length) of the vector is unchanged, the direction is flipped. Here’s the breakdown:

• The direction angle helps tell us where the vector leads.
• Flipping a vector changes its angle by \(180^{\circ}\), pointing it in the opposite direction.
• For \(\vec{B} = -\vec{A}\), the new angle is opposite to \(\vec{A}\), confirming statement "c" as correct.

Grasping the idea of direction angles allows us to visualize vector transformations, aiding in spatial reasoning and navigation applications.