Vector operations; relative position tector. Two particles...

Vector operations; relative position tector. Two particles are emitted from a common source and at a particular time have displacements: $$ \mathbf{r}_{1}=4 \hat{\mathbf{x}}+3 \hat{\mathbf{y}}+8 \hat{\mathbf{z}} \quad \mathbf{r}_{2}=2 \hat{\mathbf{x}}+10 \hat{y}+5 \hat{z} $$ (a) Sketch the positions of the particles and write the expression for the displacement $\mathbf{r}$ of particle 2 relative to particle 1 . (b) Use the scalar product to find the magnitude of each vector. $\quad$ Ans. $r_{1}=9.4 ; r_{2}=11.4 ; r=7.9$. (c) Calculate the angles between all possible pairs of the three vectors. (d) Calculate the projection of $\mathbf{r}$ on $\mathbf{r}_{1}$. Ans. $-1.2$ (e) Calculate the vector product $\mathbf{r}_{1} \times \mathbf{r}_{2}$.

Vector operations; relative position tector. Two particles are emitted from a common source and at a particular time have displacements:
$$
\mathbf{r}_{1}=4 \hat{\mathbf{x}}+3 \hat{\mathbf{y}}+8 \hat{\mathbf{z}} \quad \mathbf{r}_{2}=2 \hat{\mathbf{x}}+10 \hat{y}+5 \hat{z}
$$
(a) Sketch the positions of the particles and write the expression for the displacement $\mathbf{r}$ of particle 2 relative to particle 1 .
(b) Use the scalar product to find the magnitude of each vector. $\quad$ Ans. $r_{1}=9.4 ; r_{2}=11.4 ; r=7.9$.
(c) Calculate the angles between all possible pairs of the three vectors.
(d) Calculate the projection of $\mathbf{r}$ on $\mathbf{r}_{1}$. Ans. $-1.2$
(e) Calculate the vector product $\mathbf{r}_{1} \times \mathbf{r}_{2}$.

Key Concepts

Vector Projection

Vector projection is the process of projecting one vector onto another to determine the component of one vector in the direction of the other. This is calculated using the dot product and the magnitude of the target vector, and it is important in decomposing forces and motions along specified directions.

Angle Between Vectors

The angle between two vectors is determined using their dot product and magnitudes. It provides a measure of the directional difference between vectors and is vital in understanding geometric configurations, physical interactions, and resolving vector components in different directions.

Vector Magnitude

The vector magnitude is the length or size of a vector, calculated as the square root of the sum of the squares of its components. Magnitude is fundamental for quantifying the displacement, velocity, and force represented by vectors in physical space.

Vector (Cross) Product

The vector cross product is a binary operation on two vectors in three-dimensional space, producing a vector that is perpendicular to both input vectors. Its magnitude is equal to the area of the parallelogram formed by the original vectors, and it plays a key role in determining torques, angular momentum, and orientations in space.

Relative Position Vector

The relative position vector describes the position of one object with respect to another. It is found by subtracting the displacement vector of the reference object from the displacement vector of the other object, effectively shifting the origin to the reference object. This concept is crucial in coordinate transformation and understanding the relationship between positions in space.

Vector Displacement

Vector displacement represents the change in position of an object and is expressed as a vector quantity, which includes both magnitude and direction. It can be computed by subtracting the initial position vector from the final position vector, a process essential for determining relative positions in space.

Scalar (Dot) Product

The scalar product, also known as the dot product, is a vector operation that takes two vectors and yields a scalar. It is computed as the sum of the products of the corresponding components of the vectors, and it is used to calculate angles between vectors, projection lengths, and to determine if vectors are orthogonal.

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