Question

What angle does \(\vec{B}=\left(B_{x}, B_{y}\right)=(30.0 \mathrm{~m}, 50.0 \mathrm{~m})\) make with the positive \(x\) -axis? What angle does it make with the positive \(y\) -axis?

Short answer

Answer: The vector $\vec{B}$ makes an angle of approximately $59.04^{\circ}$ with the positive $x$-axis and approximately $149.04^{\circ}$ with the positive $y$-axis.

Step-by-step solution

Identify x and y components of the vector

The vector \(\vec{B}\) has x-component \(B_x = 30.0 m\) and y-component \(B_y = 50.0 m\).

Calculate the tangent of the angle between the vector and the positive x-axis

To find the tangent of the angle between the vector and the positive x-axis, divide the y-component by the x-component: \(\tan(\theta_x) = \frac{B_y}{B_x} = \frac{50.0}{30.0} = \frac{5}{3}\).

Calculate the angle with the positive x-axis using arctangent

To find the angle \(\theta_x\) with the positive x-axis, use the arctangent function: \(\theta_x = \arctan(\frac{5}{3})\).

Convert radians to degrees

The arctangent function typically outputs in radians. To convert radians to degrees, use the formula: \(degrees = \frac{180}{\pi} \times radians\). For \(\theta_x\), we have: \(\theta_x = \frac{180}{\pi} \times \arctan(\frac{5}{3}) \approx 59.04^{\circ}\).

Calculate the angle with the positive y-axis

The angle \(\theta_y\) between the vector and the positive y-axis is 90 degrees greater than the angle with the positive x-axis: \(\theta_y = \theta_x + 90^{\circ} = 59.04^{\circ} + 90^{\circ} \approx 149.04^{\circ}\).

Final answer

The vector \(\vec{B}\) makes an angle of approximately \(59.04^{\circ}\) with the positive \(x\)-axis and approximately \(149.04^{\circ}\) with the positive \(y\)-axis.

Key concepts

Component of a Vector

Understanding the components of a vector is crucial for analyzing physical situations in multiple dimensions. A vector is a quantity with both magnitude and direction. When we refer to the components of a vector, we're talking about its projections onto the axes of a coordinate system. For example, a vector \(\vec{V}\) in two dimensions can be represented as \(\vec{V} = (V_x, V_y)\), where \(V_x\) is the horizontal component along the x-axis, and \(V_y\) is the vertical component along the y-axis.

These components can be visualized as the legs of a right triangle, with the vector \(\vec{V}\) itself being the hypotenuse. Such decomposition is vital for simplifying calculations, as each component can be treated separately in accordance with the respective axis. This characterization is especially helpful in physics where vectors represent quantities such as displacement, velocity, and force.

Tangent Function

The tangent function is a fundamental concept in trigonometry, relating the angles of a right triangle to its side lengths. It's defined as the ratio of the opposite side to the adjacent side of an angle in a right-angled triangle. Mathematically, for an angle \(\theta\), it's written as \(\tan(\theta) = \frac{opposite}{adjacent}\).

In the context of vectors, using the tangent function allows us to find the angle that a vector makes with the positive x-axis when we know its components. The function is periodic and can have many values corresponding to the same ratio, thus the angle must be determined within the correct interval, usually between \(0^\circ\) and \(180^\circ\), to make sense in a physical context.

Arctangent Function

The arctangent function, denoted as \(\arctan\) or \(\tan^{-1}\), is the inverse of the tangent function. It's used to determine the angle whose tangent is a given number. This function comes in handy when we have the ratio of the sides of a right triangle (opposite over adjacent) and we need to find the measure of the angle.

For instance, if we know that \(\tan(\theta) = \frac{5}{3}\), we can find the angle \(\theta\) by calculating \(\theta = \arctan(\frac{5}{3})\). The output of the arctangent function is in radians, so if the problem requires an angle in degrees, further conversion is necessary. It's important to note that because of the periodic nature of the tangent function, the arctangent function can have multiple outputs, and it's restricted to values between \(\frac{-\pi}{2}\) and \(\frac{\pi}{2}\) radians to ensure it remains a function.

Radians to Degrees Conversion

Radians and degrees are two different units for measuring angles. Radians are often used in mathematics and sciences because they're based on the radius of a circle, making them a very natural way to measure angles. However, degrees are more common in everyday scenarios and certain fields of engineering.

To convert an angle from radians to degrees, we multiply the radian measure by \(\frac{180}{\pi}\). Conversely, to convert degrees to radians, we multiply the degree measure by \(\frac{\pi}{180}\). Understanding this conversion is important for interpreting angles correctly in various applications, such as when using a calculator that has trigonometric functions set to output either in radians or degrees.

Trigonometry in Physics

Trigonometry plays a pivotal role in physics as it helps us to solve problems related to forces, motion, and waves, among many other areas. The trigonometric functions, including sine, cosine, and tangent, are used to relate angles to physical quantities. In vector analysis, which is fundamental for understanding motion, trigonometry is used to calculate the direction and magnitude of vector quantities.

Physical phenomena can be resolved into perpendicular components, like forces in different directions, enabling simpler calculations and understanding of the systems at play. For example, when calculating projectile motion, trigonometry is used to find the horizontal and vertical components of the initial velocity. Trigonometry also helps us understand wave properties, such as their frequencies and amplitudes, and the relationship between electric and magnetic fields in electromagnetic waves.