Question

Express any angles using the x axis as your reference. what is the magnitude of $\vec{m}$?

Answer 1

To determine the magnitude of \(\vec{m}\) and express angles using the x-axis as the reference, specific information about the vector components is required. Once the vector components are provided, the magnitude can be calculated using the formula \(|\vec{m}| = \sqrt{m_x^2 + m_y^2}\), where \(m_x\) and \(m_y\) are the respective components of the vector.

Step-by-step explanation:

The magnitude of a vector \(\vec{m}\) in two-dimensional space, given its components \(m_x\) and \(m_y\), can be determined using the Pythagorean theorem. The formula for the magnitude (\(|\vec{m}|\)) is \(|\vec{m}| = \sqrt{m_x^2 + m_y^2}\). Here, \(m_x\) represents the horizontal component of the vector along the x-axis, and \(m_y\) represents the vertical component along the y-axis.

For example, if \(\vec{m}\) has components \
`((m_x, m_y) = (3, 4)\)`, the magnitude is calculated as
`\(|\vec{m}| = √(3^2 + 4^2) = √(9 + 16) = √(25) = 5\)`. This result represents the length or magnitude of the vector \(\vec{m}\) in the given coordinate system.

Expressing angles using the x-axis as the reference involves determining the angle \(\theta\) formed by the vector with the positive x-axis. This can be done using trigonometric functions such as
`\(\theta = \arctan\left((m_y)/(m_x)\right)\)`. The angle (0) provides the orientation of the vector in relation to the x-axis.

In conclusion, the magnitude of \(\vec{m}\) is calculated using the Pythagorean theorem, considering its horizontal and vertical components. Expressing angles with the x-axis as the reference involves using trigonometric functions to find the angle formed by the vector in the given coordinate system.

Answer 2

The x and y-components of a vector can be found using the vector's magnitude and the angle it makes with the positive x-axis. The x-component is the magnitude times the cosine of the angle, and the y-component is the magnitude times the sine of the angle.

Step-by-step explanation:

The student is asking how to express the x and y-components of a vector in terms of its magnitude, A, and direction, angle θ, with the x-axis as a reference. To find the components of a vector given its magnitude and direction, we use trigonometric relationships. If θ is the angle measured counterclockwise from the positive x-axis to the vector, the component along the x-axis (Ax) can be found by A cos(θ), while the component along the y-axis (Ay) can be found by A sin(θ). Hence, a vector with magnitude A and angle θ relative to the positive x-axis will have components Ax = A cos(θ) and Ay = A sin(θ).