In a two dimensional motion of a particle, the particle...

In a two dimensional motion of a particle, the particle moves from point $A$, position vector $\vec{r}_{1}$, to point $B$, position vector $\vec{r}_{2}$. If the magnitudes of these vectors are, respectively, $r_{1}=3$ and $r_{2}=4$ and the angles they make with the $x$ -axis are $\theta_{1}=75^{\circ}$ and $\theta_{2}=15^{\circ}$, respectively, then find the magnitude of the displacement vector.

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Key Concepts

Position Vectors

Position vectors are used to represent the location of points in space relative to an origin using a magnitude and a direction. They are fundamental in describing the coordinates of points in a plane or in space, allowing the analysis of motion and displacement.

Vector Subtraction (Displacement)

The displacement vector is obtained by subtracting the initial position vector from the final position vector. This operation indicates a change in position, reflecting both the magnitude and direction from the starting point to the end point.

Vector Components

Breaking vectors into their x and y components involves using trigonometric functions to express a vector in terms of its horizontal and vertical contributions. This approach simplifies the process of vector addition and subtraction in two-dimensional motion.

Calculation of Magnitude

The magnitude of a vector is calculated using the Pythagorean theorem once the vector is expressed in component form. This fundamental concept provides the distance or length of the displacement regardless of its directional components.

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