Question

If \(A\) and \(B\) are vectors and \(B=-A\), 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) The magnitude of \(\vec{B}\) is equal to 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) \(\vec{A}+\vec{B} = 2\vec{A}\) Answer: a) and c) are the true statements.

Step-by-step solution

Work on statement (a)

Calculate the magnitudes of \(\vec{A}\) and \(\vec{B}\) and find their relationship. Since the relationship between the vectors is \(\vec{B} = -\vec{A}\), we have: Magnitude of \(\vec{A} = \lVert\vec{A}\rVert\) Magnitude of \(\vec{B} = \lVert\vec{B}\rVert = \lVert-\vec{A}\rVert = \lVert\vec{A}\rVert\) Hence, statement (a) is true: The magnitude of \(\vec{B}\) is equal to the magnitude of \(\vec{A}\).

Work on statement (b)

Check if \(\vec{A}\) and \(\vec{B}\) are perpendicular. The dot product of two perpendicular vectors is 0. Calculate the dot product of \(\vec{A}\) and \(\vec{B}\): \(\vec{A} \cdot \vec{B} = \vec{A} \cdot (-\vec{A}) = -(\vec{A} \cdot \vec{A})\) Since neither \(\vec{A} \cdot \vec{A}\) is zero (unless \(\vec{A}\) is a zero vector which does not contain any useful information), the dot product of \(\vec{A}\) and \(\vec{B}\) is not equal to 0. Therefore, statement (b) is false: \(\vec{A}\) and \(\vec{B}\) are not perpendicular.

Work on statement (c)

Find the direction angle of \(\vec{B}\) and compare with the direction angle of \(\vec{A}\). Since \(\vec{B} = -\vec{A}\), their direction angles have a difference of \(180^{\circ}\). \(\theta_{\vec{B}} = \theta_{\vec{A}} + 180^{\circ}\) Hence, statement (c) is true: The direction angle of \(\vec{B}\) is equal to the direction angle of \(\vec{A}\) plus \(180^{\circ}\).

Work on statement (d)

Calculate the sum of vectors \(\vec{A}\) and \(\vec{B}\) and compare with \(2\vec{A}\): \(\vec{A} + \vec{B} = \vec{A} + (-\vec{A}) = \vec{0} \neq 2\vec{A}\) Thus, statement (d) is false: \(\vec{A}+\vec{B} \neq 2\vec{A}\). In conclusion, the true statements about the vectors \(\vec{A}\) and \(\vec{B}\) given that \(\vec{B} = -\vec{A}\) are (a) and (c).

Key concepts

Magnitude of a Vector

The magnitude of a vector, often denoted by vertical bars like \( \lVert \vec{A} \rVert \) for vector \(\vec{A}\), is a measure of its length or size regardless of its direction. This scalar quantity is always non-negative and can be thought of as the distance from the vector's tail to its head in a geometric representation. Understanding magnitude is fundamental in physics and engineering as it represents quantities like force, velocity, or displacement.

When looking at the problem in question, it becomes clear that the magnitude of a vector remains unchanged when the vector is multiplied by -1. This means that \( \vec{A} \) and \( \vec{B} = -\vec{A} \) have the same magnitude. This concept is crucial because it tells us that while the vectors point in opposite directions, their 'strength' or 'size' remains the same. This is represented mathematically by \( \lVert \vec{B} \rVert = \lVert -\vec{A} \rVert = \lVert \vec{A} \rVert \) confirming statement (a) in our problem.

Vector Addition and Subtraction

Vector addition and subtraction are operations that combine vectors to get a resultant vector. When adding vectors, we align the tail of the second vector to the head of the first. If instead, we want to subtract, we would invert the second vector and then add it to the first. The result is a new vector that spans from the tail of the first vector to the head of the new, adjusted second vector.

In our example, \( \vec{A} + \vec{B} \) with \( \vec{B} \) being the negative of \( \vec{A} \) effectively means subtracting \( \vec{A} \) from itself, which yields a zero vector, indicated as \( \vec{0} \) and not \( 2\vec{A} \) as suggested in statement (d). This zero vector has a magnitude of zero and does not point in any direction, which is analogous to having no movement or force applied.

Dot Product of Vectors

The dot product, also known as the scalar product, is an algebraic operation that takes two equal-length vectors and returns a single number (a scalar). The dot product of two vectors \( \vec{A} \) and \( \vec{B} \) is denoted by \( \vec{A} \cdot \vec{B} \) and is calculated as the product of their magnitudes and the cosine of the angle between them. If the dot product is zero, the vectors are perpendicular (orthogonal).

For our vectors \( \vec{A} \) and \( \vec{B} \) where \( \vec{B} = -\vec{A} \), the dot product \( \vec{A} \cdot \vec{B} \) is not zero unless \( \vec{A} \) is the zero vector which makes statement (b) false. The negative sign indicates that the two vectors are in opposite directions, but not necessarily perpendicular unless they are explicitly at a \(90^\circ\) angle.

Direction Angle of Vectors

The direction angle of a vector is the angle made with a reference axis, which is typically the positive x-axis in a two-dimensional plane. This angle helps to specify the direction of the vector in space. The direction is key to understanding how vectors combine and how they can be decomposed into components along the axes.

For vectors \( \vec{A} \) and \( \vec{B} = -\vec{A} \) as mentioned in our exercise, the direction angle of \( \vec{B} \) is \( \theta_{\vec{B}} = \theta_{\vec{A}} + 180^\circ \) which makes statement (c) true. When a vector is multiplied by -1, it is effectively rotated by \( 180^\circ \) in the plane, meaning it points in exactly the opposite direction.