Question

Sketch the vectors with the components \(\vec{A}=\left(A_{n}, A_{2}\right)=(-30.0 \mathrm{~m},\) \(-50.0 \mathrm{~m}\) ) and \(\vec{B}=\left(B_{x}, B_{y}\right)=(30.0 \mathrm{~m}, 50.0 \mathrm{~m}),\) and find the magnitudes of these vectors.

Short answer

Question: Sketch the vectors A and B with components (-30.0m, -50.0m) and (30.0m, 50.0m) respectively, and find their magnitudes. Answer: After sketching vectors A and B on a coordinate system, we can find their magnitudes using the Pythagorean theorem. The magnitudes of both vectors A and B are approximately 58.3m.

Step-by-step solution

Draw a coordinate system

First, we need to draw a coordinate system with x- and y- axes. Make sure to label the axes accordingly.

Plot the vectors

Now we will plot the two vectors \(\vec{A}\) and \(\vec{B}\) on the coordinate system using their components. The vector \(\vec{A}\) has components \((-30.0\mathrm{~m}, -50.0\mathrm{~m})\), so its endpoint is at the point \((-30, -50)\). Likewise, the vector \(\vec{B}\) has components \((30.0\mathrm{~m}, 50.0\mathrm{~m})\), so its endpoint is at the point \((30,50)\). Draw arrows from the origin to each of these points, representing the vectors.

Calculate the magnitude of vector A

To find the magnitude of vector \(\vec{A}\), we will use the Pythagorean theorem. The formula for the magnitude of a vector in two dimensions is \(|A|=\sqrt{A_x^2+A_y^2}\). Substituting the given components, the magnitude of \(\vec{A}\) is $$|A|=\sqrt{(-30.0 \mathrm{~m})^2+(-50.0 \mathrm{~m})^2}=\sqrt{900+2500}=\sqrt{3400}\mathrm{~m}\approx 58.3\mathrm{~m}$$

Calculate the magnitude of vector B

Similarly, to find the magnitude of vector \(\vec{B}\), we will again use the Pythagorean theorem. Substituting the given components, the magnitude of \(\vec{B}\) is $$|B|=\sqrt{(30.0 \mathrm{~m})^2+(50.0 \mathrm{~m})^2}=\sqrt{900+2500}=\sqrt{3400}\mathrm{~m}\approx 58.3\mathrm{~m}$$

Conclusion

We have sketched the vectors \(\vec{A}\) and \(\vec{B}\), and found their magnitudes to be \(|A|\approx 58.3\mathrm{~m}\) and \(|B|\approx 58.3\mathrm{~m}\) respectively.

Key concepts

Vector Components

Understanding vector components is crucial when dealing with vector mathematics. A vector is essentially a quantity that has both magnitude and direction. In a two-dimensional space, a vector can be split into two perpendicular components: one along the x-axis and the other along the y-axis. These components serve as the building blocks of the vector and determine its position in the coordinate system.

For instance, the vector \(\vec{A}\) with the components \(\vec{A}=(A_{x}, A_{y})=(-30.0 \mathrm{~m}, -50.0 \mathrm{~m})\) has a component of -30.0 m in the x-direction and -50.0 m in the y-direction. Effectively, each component acts as a scalar multiplier to the unit vector along the respective axis. The vector's overall influence is a combination of these two scalar quantities, representing its reach in each direction from the origin.

It's pertinent to note that negative components indicate direction opposite to the positive axes, suggesting that \(\vec{A}\) points towards the negative x and y directions.

Pythagorean Theorem

The Pythagorean theorem is an indispensable tool in vector mathematics, especially when calculating vector magnitude. Derived from the right triangle, the theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Now, when you look at the vector components as sides of a right triangle, with the vector itself as the hypotenuse, you can express the length or magnitude of the vector mathematically.

The magnitude of a two-dimensional vector \(\vec{V}\) with components \(V_{x}\) and \(V_{y}\) is given by \( |V| = \sqrt{V_x^2 + V_y^2} \). This formula is a direct application of the Pythagorean theorem and is elemental for problems involving vector magnitude calculation.

For example, the magnitude of vector \(\vec{A}\) is calculated by squaring its components, summing them, and then taking the square root, resulting in \( |A| = \sqrt{(-30)^2 + (-50)^2} \). The same process applies to vector \(\vec{B}\), highlighting the theorem's power in determining the length of vectors in a plane.

Vector Representation

Vector representation is a visual and mathematical means of conveying a vector's features. Representing vectors graphically entails plotting their components along the axes of a coordinate system to visualize their direction and relative magnitude. The x-component determines horizontal displacement from the origin, while the y-component determines vertical displacement.

When using a graphical representation, one can quickly identify the direction of the vector, with arrows pointing from the origin to the endpoint defined by the components. This process not only aids in understanding vector directionality but also sets the stage for vector addition, subtraction, and other operations by using geometrical methods.

Another aspect of vector representation is the mathematical notation, where a vector is expressed in terms of its components, as in \(\vec{A}=(A_{x}, A_{y})\). This simplifies the process of working with vectors in calculations, especially when combined with the graphical approach to gain a more comprehensive understanding of vectors in physical and theoretical contexts.