What is the angle made by $3\mathop i\limits^ \wedge + 4\mathop j\limits^ \wedge $ with the x axis?

Hint:The slope gives the measure of steepness and direction of a straight line. The slope is usually represented by the letter m. We will calculate the slope which is the tangent of the angle made by any vector or line with the x-axis.

Complete step by step answer:
The tangent of the angle made by the vector with x axis is known as the slope.Slope is the magnitude of y-axis divided by the x-axis quantity of a vector.We find the slope of the vector as,
$slope = \dfrac{4}{3}$

Thus, slope can also be called as the tangent of the angle made with the x-axis.
Let the angle be $\theta $.Thus we can say that,
$\tan \theta = \dfrac{4}{3}$
Hence the angle can be the inverse tangent of the slope. Since the tangent value is known, we can easily calculate it.
$\theta = {\tan ^{ - 1}}\dfrac{4}{3}$

Hence the angle with the x axis is ${53^ \circ }$.

Note:Any vector can be resolved into our own defined coordinates. However, for everyone’s convenience, we use a standard convention. The x axis and y axis are used to resolve the vectors. It is the unit vector coincident with the direction of x axis and j is the unit vector coincident with the direction of y axis. Similarly for 3D vectors the j is used for the unit vector coincident with the direction of z axis. Students should know the tangent values of most common angles used. Also the concept of vectors and slope must be known. Also other properties and definitions like direction ratio, direction cosine and magnitude must be known to students.