Problem 71 - Fundamentals of Physics

Question

A vector \(\vec{d}\) has a magnitude \(3.0 \mathrm{~m}\) and is directed south. What are (a) the magnitude and (b) the direction of the vector \(5.0 \vec{d}\) ? What are (c) the magnitude and (d) the direction of the vector \(-2.0 \vec{d}\) ?

Step-by-Step Solution

Answer

(a) 15.0 m, (b) south, (c) 6.0 m, (d) north.

Step 1: Finding Magnitude of Vector 5.0d

First, calculate the magnitude of the vector \(5.0 \vec{d}\). The magnitude of a scaled vector \(k\vec{d}\) is \(|k| \times \text{magnitude of } \vec{d}\). Given that \(|5.0| = 5.0\) and the magnitude of \(\vec{d}\) is \(3.0 \mathrm{~m}\), the magnitude of \(5.0 \vec{d}\) is \(5.0 \times 3.0 = 15.0 \mathrm{~m}\).

Step 2: Finding Direction of Vector 5.0d

The direction of a vector scaled by a positive number (i.e., \(5.0\) in this case) remains the same as the original vector. Since \(\vec{d}\) is directed south, \(5.0 \vec{d}\) is also directed south.

Step 3: Finding Magnitude of Vector -2.0d

Next, calculate the magnitude of the vector \(-2.0 \vec{d}\). The magnitude of \(-2.0 \vec{d}\) is \(|-2.0| \times \text{magnitude of } \vec{d}\). Since \(|-2.0| = 2.0\), the magnitude is \(2.0 \times 3.0 = 6.0 \mathrm{~m}\).

Step 4: Finding Direction of Vector -2.0d

When a vector is multiplied by a negative scalar, its direction is opposite to that of the original vector. Since \(\vec{d}\) is directed south, the direction of \(-2.0 \vec{d}\) will be opposite, i.e., north.

Key Concepts

Vector Magnitude

When we discuss vectors, one of the first things to understand is their magnitude. Simply put, the magnitude of a vector is its size or length. It's like the "distance" the vector covers. To find the magnitude of a scaled vector, you multiply the absolute value of the scaling factor by the original vector's magnitude.

• For instance, if you have a vector \( \vec{d} \) with a magnitude of \( 3.0 \mathrm{~m} \), and you want to find the magnitude of \( 5.0 \vec{d} \), you multiply the original magnitude by \( 5.0 \): \( 5.0 \times 3.0 = 15.0 \mathrm{~m} \).
• However, when the vector is multiplied by a negative scalar, the magnitude calculation still uses the absolute value of the scalar. So, if \( \vec{d} \) is scaled by \( -2.0 \), you would calculate \( |-2.0| \times 3.0 = 6.0 \mathrm{~m} \).

Understanding vector magnitude is crucial because it helps you determine how far and how much strength or force a vector represents in physical applications.

Vector Direction

Once you have the magnitude sorted, it's time to talk about direction. Direction tells us where the vector is pointed. When you multiply a vector by a positive scalar, its direction remains unchanged. The vector simply extends further in the original direction.

• For example, a vector directed south will still point south after scaling by a positive number such as \( 5.0 \).

In contrast, when you multiply a vector by a negative scalar:

• The direction flips. That means the vector will point exactly opposite to its original direction.
• So, if a vector points south and is scaled by \( -2.0 \), it will now point north.

Direction is crucial as it shows how vector quantities such as velocity and force are oriented in space.

Negative Scalar Multiplication

Negative scalar multiplication throws a unique spin on vector scaling. Unlike positive multiplication which retains the original direction, negative multiplication flips it.

• For example, multiplying vector \( \vec{d} \) (originally pointing south) by \( -2.0 \) reverses its direction to north.

During scalar multiplication:

• The magnitude is calculated using the absolute value of the scalar. Thus, \( |-2.0| \) becomes \( 2.0 \), ensuring that the vector size remains positive.

Negative scalar multiplication is useful in various scenarios like reversing force direction or changing velocity in physics.