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

<p>B = 30m</p> Signup and view all the answers

<p>Scalar multiplication</p> Signup and view all the answers

<p>Add the x-components to get the resultant x-component, and add the y-components to get the resultant y-component.</p> Signup and view all the answers

<p>Vector addition is the process of combining two or more vectors to find their resultant vector.</p> Signup and view all the answers

<p>Parallel vectors are vectors that have the same or opposite directions but may have different magnitudes.</p> Signup and view all the answers

<p>Displacement is a vector quantity that represents the change in position from an initial point to a final point.</p> Signup and view all the answers

<p>The magnitude of a vector is calculated using the Pythagorean theorem, which involves square root of the sum of the squares of its components.</p> Signup and view all the answers

<p>Vector subtraction is the process of finding the resultant vector when one vector is subtracted from another.</p> Signup and view all the answers

<p>Anti-parallel vectors are vectors that have the same magnitude but opposite directions.</p> Signup and view all the answers

<p>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 $\vec{A}$.</p> Signup and view all the answers

<p>In 2D space, the vector sum $\vec{R}$ of two vectors $\vec{A}$ and $\vec{B}$ is given by $\vec{R} = \vec{A} + \vec{B} = (A_x\hat{i} + A_y\hat{j}) + (B_x\hat{i} + B_y\hat{j}) = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}$.</p> <p>In 3D space, the vector sum $\vec{R}$ of two vectors $\vec{A}$ and $\vec{B}$ is given by $\vec{R} = \vec{A} + \vec{B} = (A_x\hat{i} + A_y\hat{j} + A_z\hat{k}) + (B_x\hat{i} + B_y\hat{j} + B_z\hat{k}) = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j} + (A_z + B_z)\hat{k}$.</p> Signup and view all the answers

<p>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.</p> <p>The scalar product is:</p> <ul> <li>Positive when $0 &lt; \theta &lt; 90$</li> <li>Negative when $90 &lt; \theta &lt; 180$</li> <li>Zero when $\theta = 90$ The scalar product is maximum when $\theta = 0$, i.e., the vectors are parallel.</li> </ul> Signup and view all the answers

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

<p>To arrange a set of vectors in order of their magnitudes, with the vector of the largest magnitude first:</p> <ol> <li>Calculate the magnitude of each vector using the formula $|\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}$.</li> <li>Sort the vectors in descending order based on their magnitudes, with the vector of the largest magnitude first.</li> </ol> Signup and view all the answers