Why vectors matter in engineering
Vectors describe quantities with both magnitude and direction. Engineering examples include force,
displacement, velocity, acceleration, moment, electric field, magnetic field, and fluid velocity.
\[
\mathbf A
=
\begin{bmatrix}
A_x\\
A_y\\
A_z
\end{bmatrix}
\]
The components \(A_x\), \(A_y\), and \(A_z\) tell how much of the vector points along each coordinate axis.
Adding two or more vectors
Vector addition is done component by component. For several vectors,
\[
\mathbf R
=
\mathbf A+\mathbf B+\mathbf C+\cdots
\]
In component form,
\[
\mathbf R
=
\begin{bmatrix}
A_x+B_x+C_x+\cdots\\
A_y+B_y+C_y+\cdots\\
A_z+B_z+C_z+\cdots
\end{bmatrix}.
\]
In a force system, \(\mathbf R\) is the resultant force. It represents the single force with the same
combined effect as all the original forces.
Subtracting two or more vectors
Vector subtraction is also done component by component. For example,
\[
\mathbf R
=
\mathbf A-\mathbf B-\mathbf C
=
\begin{bmatrix}
A_x-B_x-C_x\\
A_y-B_y-C_y\\
A_z-B_z-C_z
\end{bmatrix}.
\]
Subtraction can be interpreted as adding a reversed vector:
\[
\mathbf A-\mathbf B
=
\mathbf A+(-\mathbf B).
\]
This is useful for relative displacement, relative velocity, corrections, and comparing two vector states.
Head-to-tail construction
A geometric way to add vectors is to place the tail of each new vector at the head of the previous vector.
The resultant starts at the first tail and ends at the final head.
\[
\mathbf R
=
\mathbf v_1+\mathbf v_2+\cdots+\mathbf v_n.
\]
For subtraction, use the same construction but reverse every vector that is being subtracted.
Magnitude
The magnitude is the length of a vector. In 3D,
\[
|\mathbf A|
=
\sqrt{A_x^2+A_y^2+A_z^2}.
\]
For a force vector, this is the size of the force. For a velocity vector, it is the speed.
Unit vector
A unit vector has length \(1\). It keeps the direction of the original vector but removes its magnitude.
\[
\widehat{\mathbf A}
=
\frac{\mathbf A}{|\mathbf A|}.
\]
Unit vectors are useful when a direction is known but the actual magnitude must be assigned separately.
Dot product
The dot product combines two vectors into a scalar.
\[
\mathbf A\cdot\mathbf B
=
A_xB_x+A_yB_y+A_zB_z.
\]
It is also related to the angle between the vectors:
\[
\mathbf A\cdot\mathbf B
=
|\mathbf A|\,|\mathbf B|\cos\theta.
\]
If the dot product is positive, the vectors point mostly in the same direction. If it is negative, they point
mostly in opposite directions. If it is zero, they are perpendicular.
Projection
The projection of \(\mathbf A\) onto \(\mathbf B\) gives the part of \(\mathbf A\) that acts along
the direction of \(\mathbf B\).
\[
\operatorname{proj}_{\mathbf B}\mathbf A
=
\frac{\mathbf A\cdot\mathbf B}{|\mathbf B|^2}\mathbf B.
\]
This is important in engineering because many problems only need the component of a force or velocity along
a specific line, beam, ramp, pipe, or displacement direction.
Cross product
The cross product creates a vector perpendicular to both input vectors.
\[
\mathbf A\times\mathbf B
=
\begin{bmatrix}
A_yB_z-A_zB_y\\
A_zB_x-A_xB_z\\
A_xB_y-A_yB_x
\end{bmatrix}.
\]
Its magnitude is
\[
|\mathbf A\times\mathbf B|
=
|\mathbf A|\,|\mathbf B|\sin\theta.
\]
Cross products are used in moment and torque calculations, rotational direction, area vectors, and 3D
geometry.
Moment and torque connection
In mechanics, the moment of a force about a point is found using
\[
\mathbf M
=
\mathbf r\times\mathbf F.
\]
Here \(\mathbf r\) is the position vector from the point to the force application point, and
\(\mathbf F\) is the force vector. The direction of \(\mathbf M\) follows the right-hand rule.
Work and projection connection
Work done by a constant force along a displacement is
\[
W
=
\mathbf F\cdot\mathbf d
=
|\mathbf F|\,|\mathbf d|\cos\theta.
\]
This is a dot-product calculation. It uses only the component of the force that points along the displacement.
Common mistakes
- Adding magnitudes instead of adding components.
- Forgetting that subtraction means adding the opposite vector.
- Confusing dot product with cross product.
- Forgetting that the dot product is a scalar.
- Forgetting that the cross product is a vector.
- Projecting onto a zero vector, which is undefined.
- Ignoring units in force, velocity, displacement, and moment calculations.
