Question

Find the magnitude and direction of \(-\vec{A}+\vec{B}\), where \(\vec{A}=(23.0,59.0)\) \(\vec{B}=(90.0,-150.0)\).

Short answer

Answer: The magnitude of \(-\vec{A}+\vec{B}\) is approximately \(219.5\), and its direction is approximately \(-72.6^{\circ}\).

Step-by-step solution

Find the components of -\( \vec{A} \)

To find the components of -\( \vec{A} \), we simply change the sign of the components of \(\vec{A}\). So, in this case, the components of -\( \vec{A} \) are \((-23.0,-59.0)\).

Find the components of \(-\vec{A}+\vec{B}\)

To find the components of \(-\vec{A}+\vec{B}\), we need to add the components of \(-\vec{A}\) and \(\vec{B}\). So, we have \( x_{component} = -23.0 + 90.0 = 67.0 \) \( y_{component} = -59.0 -150.0 = -209.0 \) Thus, the components of \(-\vec{A}+\vec{B}\) are \((67.0,-209.0)\).

Find the magnitude of \(-\vec{A}+\vec{B}\)

To find the magnitude of \(-\vec{A}+\vec{B}\), we can use the Pythagorean theorem: \(|\vec{R}| = \sqrt{(x_{component})^2 + (y_{component})^2}\) \(|\vec{R}| = \sqrt{(67.0)^2 + (-209.0)^2}\) \(|\vec{R}| = \sqrt{4489.0+43681.0}\) \(|\vec{R}| = \sqrt{48170.0}\) \(|\vec{R}| \approx 219.5\) The magnitude of \(-\vec{A}+\vec{B}\) is approximately \(219.5\).

Find the direction of \(-\vec{A}+\vec{B}\)

To find the direction of \(-\vec{A}+\vec{B}\), we can use the arctangent function: \(\theta = \mathrm{atan2}(y_{component}, x_{component})\) \(\theta = \mathrm{atan2}(-209.0, 67.0)\) \(\theta \approx -1.268\) (in radians) To convert this to degrees, we can use the conversion factor: \(1 \,\mathrm{rad}\, = 180/\pi\, \mathrm{degrees}\) \(\theta \approx -1.268 \times \frac{180}{\pi}\) \(\theta \approx -72.6^{\circ}\) The direction of \(-\vec{A}+\vec{B}\) is approximately \(-72.6^{\circ}\). In conclusion, the magnitude of \(-\vec{A}+\vec{B}\) is approximately \(219.5\), and its direction is approximately \(-72.6^{\circ}\).

Key concepts

Magnitude Calculation

Understanding how to calculate the magnitude of a vector is crucial in physics, engineering, and mathematics. Magnitude represents the length or size of a vector and is denoted by the symbol '|vector|'.

When dealing with two-dimensional vectors, the magnitude can be found using the formula:
\begin{align*}|\textbf{v}| = \ \sqrt{x^2 + y^2}\,\end{align*} where \(x\) and \(y\) are the horizontal and vertical components of the vector, respectively. This is a direct application of the Pythagorean theorem, as a vector can be thought of as the hypotenuse of a right triangle, with its components being the adjacent and opposite sides.

To put this into practice using our given vector problem: the magnitude of the summed vector \(-\vec{A}+\vec{B}\) was calculated to be approximately 219.5 by first finding the components and then applying the formula for the magnitude.

Vector Components

Vector components play a pivotal role in breaking down a vector into parts that are easier to manage, often corresponding to directions such as horizontal (x-axis) and vertical (y-axis).

Each vector in two-dimensional space can be described by two components:

• The x-component, representing horizontal displacement
• The y-component, representing vertical displacement

If we have a vector \(\vec{V}=(x, y)\), its x-component is 'x' and its y-component is 'y'.

In the context of our exercise: the vector \(-\vec{A}\) was found by negating the components of \(\vec{A}\), resulting in \((-23.0, -59.0)\). Adding this to \(\vec{B}\), we found the resulting vector's components to be (67.0, -209.0), which are essential for the subsequent magnitude and direction calculations.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, stating that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The theorem is represented by the equation: \(a^2 + b^2 = c^2\), where 'c' is the hypotenuse.

In the field of vectors, this theorem helps us calculate the magnitude of a vector, which is equivalent to finding the length of the hypotenuse in a right triangle formed by the vector's components. Applied to our problem, we calculated the magnitude of the vector resulting from \(-\vec{A}+\vec{B}\) by squaring its components and taking the square root of their sum, as outlined in the magnitude calculation.

Arctangent Function

The arctangent function is the inverse of the tangent function and is used to find an angle whose tangent is a given number. It is commonly abbreviated as 'atan' or 'tan^-1'. The function is especially useful in vector mathematics for determining the direction or angle of a vector from its components.

When using the arctangent function, one must consider the signs of the vector components to determine the correct quadrant for the angle. For better accuracy, the 'atan2' variation is used, which takes into account both x and y components, providing an angle between \(-\pi\) and \(\pi\).

In our problem, we calculated the direction of the summed vector \(-\vec{A}+\vec{B}\) using the 'atan2' function with the components (67.0, -209.0), resulting in an angle of approximately -72.6°, accounting for the correct orientation of the vector.