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Chapter 4: Problem 35

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 \(-2.0 \vec{d}\) ?

Short Answer

Expert verified
Magnitudes: 15.0 m (5.0d), 6.0 m (-2.0d); Directions: South (5.0d), North (-2.0d).

Step by step solution

01

- Magnitude of Vector Multiplication (5.0d)

To find the magnitude of the vector after it has been multiplied by 5.0, multiply the original magnitude by 5.0. Given the original magnitude is 3.0 m, the calculation is: \[ \text{Magnitude} = 5.0 \times 3.0 \mathrm{~m} = 15.0 \mathrm{~m} \]
02

- Direction of Vector (5.0d)

The direction of a vector remains unchanged when it is multiplied by a positive scalar. Since the original vector \( \vec{d} \) is directed south, \( 5.0 \vec{d} \) is also directed south.
03

- Magnitude of Vector Multiplication (-2.0d)

To find the magnitude of the vector after it has been multiplied by -2.0, take the absolute value of the multiplication. Given the original magnitude is 3.0 m, the calculation is: \[ \text{Magnitude} = |-2.0 \times 3.0 \mathrm{~m}| = 6.0 \mathrm{~m} \]
04

- Direction of Vector (-2.0d)

When a vector is multiplied by a negative scalar, its direction is reversed. Since the original vector \( \vec{d} \) is directed south, the vector \( -2.0 \vec{d} \) is directed north.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

vector magnitude
The magnitude of a vector gives information about its size or length, regardless of its direction. You can think of it as the 'how much' part of the vector.
The magnitude is typically represented as \( \|\vec{d}\| \). For example, if a vector has a magnitude of 3.0 meters, it means the vector has this length in space.
When a vector is multiplied by a scalar, its magnitude changes. However, only the number part changes, not the direction part.
  • Multiplying by a positive scalar just changes the length. For instance, \( 5.0 \vec{d} \) means you multiply the original magnitude 3.0 meters by 5, giving you 15.0 meters.
  • For a negative scalar, you still change the magnitude to positive but also reverse the direction. For example, \( -2.0 \vec{d} \): magnitude becomes 6.0 meters, and the direction reverses.
vector direction
The direction of a vector indicates the way it points in space.
It is usually represented as an angle or a cardinal direction (e.g., north, south, east, west). A vector's direction is vital because it complements its magnitude to describe the vector fully.
When multiplying a vector by a positive scalar, the direction remains the same. For example, if a vector \( \vec{d} \) is directed south, then \( 5.0 \vec{d} \) is also directed south.
However, multiplying by a negative scalar reverses its direction. So, if \( \vec{d} \) is directed south, then \( -2.0 \vec{d} \) will point the opposite way, which is north.
It is essential to note that while the magnitude changes based on the scalar, the direction either stays the same or flips, depending on the scalar's sign.
scalar multiplication
Scalar multiplication involves multiplying a vector by a scalar (a number). This operation transforms the vector鈥檚 magnitude and sometimes its direction.
The rule is straightforward:
  • Multiply the magnitude with the scalar.
  • Keep the same direction if the scalar is positive.
  • Reverse the direction if the scalar is negative.
In the exercise, multiplying by 5.0 results in \( 5.0 \vec{d} \), where the magnitude becomes 15.0 meters and the direction remains south. When multiplying by \( -2.0 \vec{d} \), the magnitude becomes 6.0 meters, but the direction flips to north.
To summarize:
  • Positive scalar: magnitude changes, direction stays the same.
  • Negative scalar: magnitude changes, direction flips.
Understanding scalar multiplication helps in simplifying and predicting the effects on vectors in physics and engineering problems.

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Most popular questions from this chapter

What is the sum of the following four vectors in (a) unit-vector notation and (b) magnitude-angle notation? For the latter, give the angle in both degrees and radians. Positive angles are counterclockwise from the positive direction of the \(x\) axis; negative angles are clockwise. $$ \begin{array}{ll} \vec{E}: 6.00 \mathrm{~m} \text { at }+0.900 \mathrm{rad} & \vec{F}: 5.00 \mathrm{~m} \text { at }-75.0^{\circ} \\ \vec{G}: 4.00 \mathrm{~m} \text { at }+1.20 \mathrm{rad} & \vec{H}: 6.00 \mathrm{~m} \text { at }-210^{\circ} \end{array} $$

A protester carries his sign of protest \(40 \mathrm{~m}\) along a straight path, then \(20 \mathrm{~m}\) along a perpendicular path to his left, and then \(25 \mathrm{~m}\) up a water tower. (a) Choose and describe a coordinate system for this motion. In terms of that system and in unitvector notation, what is the displacement of the sign from start to end? (b) The sign then falls to the foot of the tower. What is the magnitude of the displacement of the sign from start to this new end?

What are (a) the \(x\) -component and (b) the \(y\) -component of a vector \(\vec{a}\) in the \(x y\) plane if its direction is \(250^{\circ}\) counterclockwise from the positive direction of the \(x\) axis and its magnitude is \(7.3 \mathrm{~m}\) ?

Consider how the components of a vector in the plane change if I change the reference point. Suppose I start with a coordinate system with an origin at \(O\). An arbitrary vector \(\vec{r}=x \hat{\mathrm{i}}+y \hat{\mathrm{j}}\) with coordinates \((x, y)\) specifies a point in this system. Suppose also that I have another point \(O^{\prime}\) specified in this coordinate system by a vector \(\vec{A}=A_{x} \hat{\mathrm{i}}+A_{y} \hat{\mathrm{j}}\). If I change my origin to \(O^{\prime}\) (without rotating the axes), what would the coordinates be for the point specified by \(\vec{r}\) ?

A vector \(\vec{B}\), with a magnitude of \(8.0 \mathrm{~m}\), is added to a vector \(\vec{A}\), which lies along an \(x\) axis. The sum of these two vectors is a third vector that lies along the \(y\) axis and has a magnitude that is twice the magnitude of \(\vec{A}\). What is the magnitude of \(\vec{A}\) ?
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