Question

Suppose a vector$\overrightarrow{\mathbf{V}}$makes an angle$\varphi$with respect to the y-axis. What could be the xand ycomponents of vector$\overrightarrow{\mathbf{V}}$?

Short answer

Thexand ycomponents of vector $\overrightarrow{\mathbf{V}}$are $V\sin \varphi$and $V\cos \varphi$, respectively.

Step-by-step solution

Step 1. Significance of the vector components

A vector can be divided into two or three components depending upon the coordinate axes. The vector can be split into two or three scalar forms of quantities.

The components of the vector can be generally obtained when the vector is inclined at some angle with any of the coordinate’s axes.

The resultant of two or more vectors can be calculated with the help of the addition or the subtraction of the horizontal and vertical components of the individual vectors.

Step 2. Procedure for the determination of the components of vector V→

Consider that the Cartesian coordinates system consists of xand y-axes. Vector $\overrightarrow{\mathbf{V}}$is making an angle of$\varphi$with the y-axis. Then, vector $\overrightarrow{\mathbf{V}}$is broken into two components along the x and y-axes.

While resolving any of the vectors, it can be noted that the axis with which the vector is inclined is considered the (cosine) component of the same vector. The other axis could be considered the (sine) component of the same vector.

It is given that vector $\overrightarrow{\mathbf{V}}$is inclined with the y-axis. So, the y-axis consists of the (cosine) component, and therefore, the other axis, that is, the x-axis, can be taken as the (sine) component of $\overrightarrow{\mathbf{V}}$vector.

Step 3. The determination of the components of vector V→

The x-component of vector $\overrightarrow{\mathbf{V}}$can be expressed as

$V_x=V\sin \varphi$.

Here, $V_x$is the x-component of the $\overrightarrow{\mathbf{V}}$, $\varphi$ is the angle of inclination of the vector with the y-axis, and V is the magnitude of the vector.

The y-component of vector $\overrightarrow{\mathbf{V}}$can be expressed as

$V_y=V\cos \varphi$.

Here, $V_y$is the y-component of the vector $\overrightarrow{\mathbf{V}}$, and$\varphi$ is the angle of inclination of the vector with the y-axis.

Thus, thexand ycomponents of vector $\overrightarrow{\mathbf{V}}$are $V\sin \varphi$and $V\cos \varphi$, respectively.