Question

The resultant of the two-dimensional vectors \((1.5 \mathrm{~m}, 0.7 \mathrm{~m})\) \((-3.2 \mathrm{~m}, 1.7 \mathrm{~m}),\) and \((1.2 \mathrm{~m},-3.3 \mathrm{~m})\) lies in quadrant a) I b) II c) III d) IV

Short answer

a) I b) II c) III d) IV Answer: c) III

Step-by-step solution

Add the vectors

Add the three given vectors component-wise: \((1.5, 0.7) + (-3.2, 1.7) + (1.2, -3.3)\). To do this, we add their x-components and y-components together, separately.

Calculate x-component of the resultant

Add the x-components of the given vectors: \(1.5 - 3.2 + 1.2 = -0.5~\mathrm{m}\).

Calculate y-component of the resultant

Add the y-components of the given vectors: \(0.7 + 1.7 - 3.3 = -0.9~\mathrm{m}\).

Determine the quadrant

Now we have the resultant vector (-0.5, -0.9). The x-component is negative, and the y-component is negative, which means the resultant vector lies in Quadrant III. Therefore, the correct answer is: c) III

Key concepts

Two-Dimensional Vectors

Two-dimensional vectors are mathematical objects with both magnitude and direction that are confined to a plane. They are fundamental in physics as they describe quantities that have both size and direction in a two-dimensional space, such as force, velocity, or displacement.

Imagine walking along a grid on a flat surface: starting from a point, you could move right and upwards, or left and downwards, or any combination of the two axes. This is what we describe with two-dimensional vectors which typically have two components: one for each axis of our grid, which we refer to as the x-axis (horizontal) and y-axis (vertical).

In vector addition, like the type we see in this textbook exercise, vectors are added together by summing their respective components. This process is comparable to walking a certain distance in one direction and then walking again in another direction. The final point you reach can be found by adding together each separate movement in the x-direction and the y-direction.

Resultant Vector

The resultant vector is essentially the sum of two or more vectors. It represents the combined effect of the different vectors acting together. Think of it as the final destination in a treasure map that has you move different steps in specific directions; the spot where you find the treasure is the resultant of those movements.

For example, if you have three separate forces acting on an object, the resultant force would be the single force that could replace these three and have the same effect on the object. Using the components we've previously calculated, the resultant vector can be determined, and from this, the direction and magnitude of the combined vector can be inferred.

The resultant vector is particularly important in physics to understand the overall effect of multiple vector quantities acting at a point. In real-world terms, if you're pushing a shopping cart forwards and your friend pushes it to the right, the resultant vector would describe the cart's actual path.

Vector Components

Vector components are essentially the projections of a vector onto the axes of the coordinate system used (usually the x and y axes). Breaking down a vector into its components is like finding out how much of that vector is pointing in the direction of each axis.

This is similar to deciphering a team's success in a game by looking at individual players' performances. The components tell us how much of the vector quantity is in the horizontal direction, and how much is in the vertical direction. In the case of our exercise, we took the horizontal and vertical components (x and y, respectively) of each vector and added them separately to get the respective components of the resultant vector.

Exercise Improvement Advice

• Understand that the components of a vector are its 'building blocks' and act independently of each other.
• Adding vectors together is like adding the individual horizontal and vertical distances travelled.
• Just as you'd separate eggs and flour when baking a cake before mixing, vector components must first be dealt with individually before combining to form the resultant vector.

Knowing how to work with vector components is crucial for simplifying complex problems in physics into more manageable parts.