Vector Addition and Unit Vectors Quiz

Dx = D * cos(α), Dy = D * sin(α)

B = sqrt(Bx^2 + By^2), Ɵ = arctan(By/Bx)

Bx = 3Ax, By = 3Ay

B = 30m

Scalar multiplication

Add the x-components to get the resultant x-component, and add the y-components to get the resultant y-component.

Vector addition is the process of combining two or more vectors to find their resultant vector.

Parallel vectors are vectors that have the same or opposite directions but may have different magnitudes.

Displacement is a vector quantity that represents the change in position from an initial point to a final point.

The magnitude of a vector is calculated using the Pythagorean theorem, which involves square root of the sum of the squares of its components.

Vector subtraction is the process of finding the resultant vector when one vector is subtracted from another.

Anti-parallel vectors are vectors that have the same magnitude but opposite directions.

A unit vector is a vector that has a magnitude of 1, with no units. Its only purpose is to pointthat is, to describe a direction in space. In contrast, a vector with magnitude has both a direction and a magnitude (size). The relationship between a vector with magnitude $A$ and its corresponding unit vector $\hat{A}$ is given by $\vec{A} = A\hat{A}$, where $\hat{A}$ is the unit vector in the direction of $\vec{A}$.

In 2D space, the vector sum $\vec{R}$ of two vectors $\vec{A}$ and $\vec{B}$ is given by $\vec{R} = \vec{A} + \vec{B} = (A_x\hat{i} + A_y\hat{j}) + (B_x\hat{i} + B_y\hat{j}) = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}$.

In 3D space, the vector sum $\vec{R}$ of two vectors $\vec{A}$ and $\vec{B}$ is given by $\vec{R} = \vec{A} + \vec{B} = (A_x\hat{i} + A_y\hat{j} + A_z\hat{k}) + (B_x\hat{i} + B_y\hat{j} + B_z\hat{k}) = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j} + (A_z + B_z)\hat{k}$.

The scalar product (dot product) of two vectors $\vec{A}$ and $\vec{B}$ is defined as $\vec{A} \cdot \vec{B} = AB\cos\theta$, where $A$ and $B$ are the magnitudes of the vectors and $\theta$ is the angle between them.

The scalar product is:

• Positive when $0 < \theta < 90$
• Negative when $90 < \theta < 180$
• Zero when $\theta = 90$ The scalar product is maximum when $\theta = 0$, i.e., the vectors are parallel.

The vector product (cross product) of two vectors $\vec{A}$ and $\vec{B}$ is defined as $\vec{C} = \vec{A} \times \vec{B} = (A_yB_z - A_zB_y)\hat{i} + (A_zB_x - A_xB_z)\hat{j} + (A_xB_y - A_yB_x)\hat{k}$.

The direction of the resulting vector $\vec{C}$ is given by the right-hand rule: if the fingers of the right hand are curled in the direction from $\vec{A}$ to $\vec{B}$, then the thumb will point in the direction of $\vec{C}$.

To arrange a set of vectors in order of their magnitudes, with the vector of the largest magnitude first:

1. Calculate the magnitude of each vector using the formula $|\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}$.
2. Sort the vectors in descending order based on their magnitudes, with the vector of the largest magnitude first.