1 Vectors for Engineering Mechanics Free Online FlashCards

What distinguishes a vector from a scalar?

A vector has both magnitude and direction. Force, displacement, and moment are vectors; mass and time are scalars.

Why specify coordinate axes and sign convention?

A vector’s components depend on the coordinate axes chosen. State the axes and sign convention before calculating.

What directions do $\mathbf{i},\mathbf{j},\mathbf{k}$ represent?

In a right-handed Cartesian system, $\mathbf{i},\mathbf{j},\mathbf{k}$ point along the positive $x,y,z$ axes, respectively, and each has magnitude $1$.

How do scalar and vector components differ?

In $\vec A=A_x\mathbf{i}+A_y\mathbf{j}+A_z\mathbf{k}$, each scalar component is a signed number; each vector component, such as $A_x\mathbf{i}$, includes direction.

What does a negative vector component indicate?

A negative scalar component points opposite the positive direction of its coordinate axis.

How are planar vector components found from magnitude and angle?

For a vector of magnitude $A$ at angle $\theta$ counterclockwise from the positive $x$-axis, $A_x=A\cos\theta$ and $A_y=A\sin\theta$.

How do you recover a 3D vector’s magnitude from its components?

In three dimensions, $|\vec A|=\sqrt{A_x^2+A_y^2+A_z^2}$.

How do you find the vector from point $P$ to point $Q$?

Subtract the tail coordinates from the head coordinates: $\overrightarrow{PQ}=(x_Q-x_P)\mathbf{i}+(y_Q-y_P)\mathbf{j}+(z_Q-z_P)\mathbf{k}$.

How do you obtain a unit vector in the direction of $\vec A$?

For nonzero $\vec A$, its direction unit vector is $\hat{\mathbf{u}}_A=\frac{\vec A}{|\vec A|}$, which has magnitude $1$.

How are vectors added in component form?

Add corresponding components algebraically: $\vec A+\vec B=(A_x+B_x)\mathbf{i}+(A_y+B_y)\mathbf{j}+(A_z+B_z)\mathbf{k}$.

How does scalar multiplication affect a vector?

For scalar $c$, $c\vec A$ has magnitude $|c||\vec A|$. Positive $c$ preserves direction; negative $c$ reverses it.

What does the dot product return, and how is it calculated?

The dot product is a scalar: $\vec A\cdot\vec B=A_xB_x+A_yB_y+A_zB_z=|\vec A||\vec B|\cos\phi$, where $\phi$ is the angle between the vectors.