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 $\mathbf{r}_{1}$ to point $B$, position vector $\mathbf{r}_{2}$. 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 $15^{\circ}$, respectively, then find the magnitude of the displacement vector (a) 15 (b) $\sqrt{13}$ (c) 17 (d) $\sqrt{15}$

Key Concepts

Magnitude of a Vector

The magnitude of a vector is a measure of its length or size. It is calculated using the Euclidean norm, which in two dimensions is found by taking the square root of the sum of the squares of its components.

Vector Subtraction

Vector subtraction is the process used to find the difference between two vectors. It involves subtracting corresponding components of the vectors and results in a new vector that represents the change or relative position between the original vectors.

Law of Cosines in Vector Analysis

The law of cosines can be applied in vector analysis to determine the magnitude of the result of vector subtraction when the angle between the original vectors is known. This method is particularly useful when the vectors are defined in polar form, as it relates the lengths of the vectors and the cosine of the angle between them.

Position Vectors

Position vectors are used to represent the location of points in space with respect to an origin. They are essential in describing the initial, final, or any specific positions in a coordinate system, which allows for a clear analysis of spatial relationships.

Displacement Vector

The displacement vector represents the change in position of an object, obtained by subtracting its initial position vector from its final position vector. It encapsulates both the distance and the direction of movement between two points.

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