Question

Consider two displacements, one of magnitude \(3 \mathrm{~m}\) and another of magnitude \(4 \mathrm{~m}\). Show how the displacement vectors may be combined to get a resultant displacement of magnitude \((\) a) \(7 \mathrm{~m},(b) 1 \mathrm{~m}\), and \((c) 5 \mathrm{~m}\)

Short answer

The vectors can be arranged in three different ways to get the desired results. For a resultant displacement of 7m, the vectors of 3m and 4m must be arranged in the same direction. For a resultant displacement of 1m, the vectors must be directed opposite each other. For a resultant displacement of 5m, the vectors should be arranged at right angles to each other.

Step-by-step solution

Solving for 7m magnitude resultant

Place the two vectors end to end in the same direction, starting from the same point. When two vectors are added together, the resulting vector, or the resultant, simply combines the magnitudes of the two original vectors. In this case, \(3 m + 4 m = 7 m\). Therefore, aligning the displacements in the same direction will result in a resultant displacement of \(7 m\).

Solving for 1m magnitude resultant

Proceed in a similar manner, but now place the two vectors in opposite directions, starting from the same point. With the vectors arranged in this manner, the magnitudes subtract from each other instead of add together, \(4 m - 3 m = 1 m\). Therefore, arranging the displacements opposite to one another will result in a resultant displacement of \(1 m\).

Solving for 5m magnitude resultant

Arranging two vectors at right angles to each other will add the squares of their magnitudes, the square root of the sum will be the magnitude of the resultant. Therefore the magnitude of the resultant, R, can be given by \(R = \sqrt{(3^2 + 4^2)} = 5 m\). Thus, a displacement of \(5m\) will be obtained when the given vectors are arranged at right angles to each other.

Key concepts

Understanding Resultant Displacement

Resultant displacement refers to the single vector that combines the effects of multiple individual vectors. It is the vector obtained when two or more vectors are added together. Think of it as the end result of moving in several directions. To find the resultant displacement, we often use vector addition, which is simply like following a series of directions on a map.

When dealing with displacement vectors, the magnitude and direction are crucial. By arranging two vectors in the same direction, you simply add their magnitudes. For example, two vectors of magnitudes 3 m and 4 m placed one after the other yield a resultant of 7 m.

• Place both vectors in the same direction to maximize the resultant.
• Align them oppositely to minimize the resultant.
• To achieve a diagonal between them, place them at right angles.

Thus, the way vectors are positioned entirely governs the resultant displacement.

Calculating the Magnitude of Vectors

The magnitude of a vector is a measure of its length or size and is always a positive value. Depending on how vectors are positioned relative to each other, the combined magnitude will differ. For instance, when two vectors are aligned in the same direction, their magnitudes add up directly.Consider our example: by placing vectors of 3 m and 4 m in opposite directions, the magnitudes effectively subtract, giving a resultant of 1 m. This subtraction represents the net effect of opposing directions. Hence, finding the magnitude involves considering both the length of the vectors and their direction relative to each other.

• For opposite vectors, subtract the smaller from the larger.
• For vectors at right angles, use the Pythagorean theorem to determine the resultant.

Positioning vectors orthogonally results in a magnitude found using the formula: \[ R = \sqrt{(3^2 + 4^2)} = 5 \text{ m} \]. This follows from the concept of right-angle triangle sides contributing to the hypotenuse.

Exploring Vector Direction

Direction is an essential aspect of understanding vectors. Its importance becomes evident when realizing how it determines the resultant in vector addition. While the magnitude tells us how big or long a vector is, direction tells us where it points. It's crucial when vectors with identical magnitudes can lead to different resultant displacements because they are oriented differently. When arranging vectors:

• If vectors point in the same direction, the resultant is as straight and long as possible.
• Opposing directions reduce the resultant to the difference between magnitudes.
• Right-angle arrangements give a diagonal resultant whose magnitude can be found using mathematical principles like the Pythagorean theorem.

Vector direction therefore not only defines the physical path but is also a vital component when performing calculations, including adding and subtracting vectors to determine resultant displacement.