Problem 58 - Fundamentals of Physics

Question

A vector \(\vec{d}\) has a magnitude of \(2.5 \mathrm{~m}\) and points north. What are (a) the magnitude and (b) the direction of \(4.0 \vec{d}\) ? What are (c) the magnitude and (d) the direction of \(-3.0 \vec{d}\) ?

Step-by-Step Solution

Answer

(a) 10.0 m, (b) north, (c) 7.5 m, (d) south.

Step 1: Understanding the Original Vector

We have a vector \(\vec{d}\) with a magnitude of \(2.5 \, \text{m}\) and it points north. This information tells us the vector's size and direction.

Step 2: Calculating Magnitude of 4.0d

To find the magnitude of the vector \(4.0 \vec{d}\), multiply the original magnitude by 4: \(4.0 \times 2.5 = 10.0 \, \text{m}\).

Step 3: Determining Direction of 4.0d

Since \(4.0 \vec{d}\) is a scalar multiple of \(\vec{d}\), it has the same direction as \(\vec{d}\), which is north.

Step 4: Calculating Magnitude of -3.0d

For the vector \(-3.0 \vec{d}\), multiply the original magnitude by 3: \(-3.0 \times 2.5 = -7.5 \, \text{m}\). However, magnitude is always positive, so it's \(7.5 \, \text{m}\).

Step 5: Determining Direction of -3.0d

The vector \(-3.0 \vec{d}\) points in the opposite direction of \(\vec{d}\). Since \(\vec{d}\) points north, \(-3.0 \vec{d}\) points south.

Key Concepts

Vector Magnitude

The vector magnitude represents the length or size of the vector, which is always a non-negative value. To calculate the magnitude of a vector that has been multiplied by a scalar, you simply multiply the original magnitude by the absolute value of the scalar. For instance, if a vector \(\vec{d}\) has a magnitude of \(2.5 \, \mathrm{m}\), multiplying it by \(4.0\) gives a new vector magnitude:

• \(4.0 \times 2.5 = 10.0 \, \mathrm{m}\)

Similarly, when multiplying by \(-3.0\), the negative sign only affects direction, so the magnitude remains:

• \(|-3.0| \times 2.5 = 7.5 \, \mathrm{m}\)

Always remember, magnitude does not have a direction, and it is always expressed as a positive number.

Vector Direction

Vector direction indicates where the vector is pointing. It's important to retain this concept during any vector operation. In physics and mathematics, each vector's direction is as crucial as its magnitude. When dealing with scalar multiplication, it's vital to note that the direction remains unchanged if the scalar is positive. For instance:

• The vector \(4.0 \vec{d}\) maintains the original direction of \(\vec{d}\), which is north.

Direction can be defined by angles, but in straightforward cases like our example, cardinal directions can be used for clarity. Therefore, understanding and visualizing direction helps you comprehend how the vector behaves in space.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar, which is a simple number. This operation scales the vector without affecting its direction unless the scalar is negative. The formula is simple:

• Magnitude of resultant vector \(= |\text{scalar}| \times \text{original magnitude}\)

For instance, multiplying a vector \(\vec{d}\) with magnitude \(2.5 \, \mathrm{m}\) by \(4.0\) results in:

• Magnitude: \(10.0 \, \mathrm{m}\)
• Direction: same as \(\vec{d}\)

When the scalar is negative, the result flips the vector direction, which leads us to our next concept. Scalar multiplication remains a fundamental operation in vector mathematics because it modifies vector properties efficiently.

Opposite Direction of Vectors

When a vector is multiplied by a negative scalar, the result is a vector pointing in the opposite direction. This concept of opposite direction is crucial for understanding vector scenarios in real life, like navigational tasks or simulations. For example, multiplying the vector \(\vec{d}\) by \(-3.0\) affects it this way:

• The magnitude becomes positive \(7.5 \, \mathrm{m}\) but the direction reverses from north to south.

The essence of opposite direction applications is in changing the course while keeping the size intact. Remember, this transformation only affects the pointing direction, not the magnitude itself, adding flexibility to vector manipulation.