Vector Operations

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Vector Operations

Vectors describe quantities that have both magnitude and direction. A two-dimensional vector can be written as $\mathbf{v}=\langle v_x,v_y\rangle$, and a three-dimensional vector can be written as $\mathbf{v}=\langle v_x,v_y,v_z\rangle$.

Magnitude of a Vector

The magnitude of a vector measures its length. For $\mathbf{v}=\langle v_x,v_y\rangle$, the magnitude is:

$$\lVert\mathbf{v}\rVert=\sqrt{v_x^2+v_y^2}.$$

For a three-dimensional vector, the magnitude includes the $z$-component:

$$\lVert\mathbf{v}\rVert=\sqrt{v_x^2+v_y^2+v_z^2}.$$

For example, if $\mathbf{v}=\langle 3,4\rangle$, then:

$$\lVert\mathbf{v}\rVert=\sqrt{3^2+4^2}=\sqrt{25}=5.$$

Direction of a Vector

The direction angle of a two-dimensional vector can be found from its components:

$$\theta=\operatorname{atan2}(v_y,v_x).$$

For $\mathbf{v}=\langle 3,4\rangle$, the direction angle is:

$$\theta=\tan^{-1}\left(\frac{4}{3}\right)\approx53.1^\circ.$$

Vector Addition

Vectors are added component by component. If $\mathbf{a}=\langle a_x,a_y\rangle$ and $\mathbf{b}=\langle b_x,b_y\rangle$, then:

$$\mathbf{a}+\mathbf{b}=\langle a_x+b_x,\;a_y+b_y\rangle.$$

For example, if $\mathbf{a}=\langle2,5\rangle$ and $\mathbf{b}=\langle4,-1\rangle$, then:

$$\mathbf{a}+\mathbf{b}=\langle2+4,\;5-1\rangle=\langle6,4\rangle.$$

Vector Subtraction

Vector subtraction is also performed component by component:

$$\mathbf{a}-\mathbf{b}=\langle a_x-b_x,\;a_y-b_y\rangle.$$

This can also be understood as adding the opposite of the second vector:

$$\mathbf{a}-\mathbf{b}=\mathbf{a}+(-\mathbf{b}).$$

Scalar Multiplication

Multiplying a vector by a scalar changes its magnitude and may reverse its direction if the scalar is negative. For scalar $k$ and vector $\mathbf{v}$:

$$k\mathbf{v}=\langle kv_x,\;kv_y\rangle.$$

If $\mathbf{v}=\langle3,-2\rangle$, then:

$$2\mathbf{v}=\langle6,-4\rangle.$$

Unit Vectors

A unit vector has magnitude $1$. The unit vector in the direction of a nonzero vector $\mathbf{v}$ is:

$$\widehat{\mathbf{v}}=\frac{\mathbf{v}}{\lVert\mathbf{v}\rVert}.$$

For $\mathbf{v}=\langle3,4\rangle$, the unit vector is:

$$\widehat{\mathbf{v}}=\left\langle\frac35,\frac45\right\rangle.$$

The standard Cartesian unit vectors are:

$$\mathbf{i}=\langle1,0,0\rangle,\qquad\mathbf{j}=\langle0,1,0\rangle,\qquad\mathbf{k}=\langle0,0,1\rangle.$$

Using these unit vectors, a three-dimensional vector can be written as:

$$\mathbf{v}=v_x\mathbf{i}+v_y\mathbf{j}+v_z\mathbf{k}.$$

Resolving a Vector Into Components

If a vector has magnitude $V$ and direction angle $\theta$, its rectangular components are:

$$V_x=V\cos\theta.$$

$$V_y=V\sin\theta.$$

For a force of $100\text{ N}$ acting at $30^\circ$ above the horizontal:

$$F_x=100\cos30^\circ\approx86.6\text{ N}.$$

$$F_y=100\sin30^\circ=50\text{ N}.$$

Therefore, the force vector is:

$$\mathbf{F}=\langle86.6,50\rangle\text{ N}.$$

Resultant Vectors

The resultant vector is the sum of two or more vectors:

$$\mathbf{R}=\sum_{i=1}^{n}\mathbf{v}_i.$$

In component form:

$$R_x=\sum_{i=1}^{n}v_{x,i},\qquad R_y=\sum_{i=1}^{n}v_{y,i}.$$

The magnitude of the resultant is:

$$\lVert\mathbf{R}\rVert=\sqrt{R_x^2+R_y^2}.$$

The direction of the resultant can be found using:

$$\theta_R=\operatorname{atan2}(R_y,R_x).$$

Dot Product

The dot product combines two vectors to produce a scalar. For vectors in three dimensions:

$$\mathbf{a}\cdot\mathbf{b}=a_xb_x+a_yb_y+a_zb_z.$$

The dot product can also be written in terms of the angle between the vectors:

$$\mathbf{a}\cdot\mathbf{b}=\lVert\mathbf{a}\rVert\lVert\mathbf{b}\rVert\cos\theta.$$

For $\mathbf{a}=\langle2,3\rangle$ and $\mathbf{b}=\langle4,-1\rangle$:

$$\mathbf{a}\cdot\mathbf{b}=(2)(4)+(3)(-1)=5.$$

Two nonzero vectors are perpendicular when their dot product is zero:

$$\mathbf{a}\cdot\mathbf{b}=0.$$

Angle Between Two Vectors

For two nonzero vectors, the angle between them is:

$$\theta=\cos^{-1}\left(\frac{\mathbf{a}\cdot\mathbf{b}}{\lVert\mathbf{a}\rVert\lVert\mathbf{b}\rVert}\right).$$

A positive dot product indicates an acute angle. A zero dot product indicates a right angle. A negative dot product indicates an obtuse angle.

Vector Projection

The scalar projection of $\mathbf{a}$ onto $\mathbf{b}$ is:

$$\operatorname{comp}_{\mathbf{b}}\mathbf{a}=\frac{\mathbf{a}\cdot\mathbf{b}}{\lVert\mathbf{b}\rVert}.$$

The vector projection of $\mathbf{a}$ onto $\mathbf{b}$ is:

$$\operatorname{proj}_{\mathbf{b}}\mathbf{a}=\frac{\mathbf{a}\cdot\mathbf{b}}{\lVert\mathbf{b}\rVert^2}\mathbf{b}.$$

Cross Product

The cross product applies to three-dimensional vectors and produces a vector perpendicular to the two original vectors:

$$\mathbf{a}\times\mathbf{b}=\begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k}\\a_x & a_y & a_z\\b_x & b_y & b_z\end{vmatrix}.$$

Expanding the determinant gives:

$$\mathbf{a}\times\mathbf{b}=\left\langle a_yb_z-a_zb_y,\;a_zb_x-a_xb_z,\;a_xb_y-a_yb_x\right\rangle.$$

The magnitude of the cross product is:

$$\lVert\mathbf{a}\times\mathbf{b}\rVert=\lVert\mathbf{a}\rVert\lVert\mathbf{b}\rVert\sin\theta.$$

This magnitude is also the area of the parallelogram formed by the two vectors:

$$A_{\text{parallelogram}}=\lVert\mathbf{a}\times\mathbf{b}\rVert.$$

The area of the corresponding triangle is half that value:

$$A_{\triangle}=\frac12\lVert\mathbf{a}\times\mathbf{b}\rVert.$$

Equilibrium of a Particle

A particle is in equilibrium when the sum of all forces acting on it is zero:

$$\sum\mathbf{F}=\mathbf{0}.$$

In two dimensions, this gives two scalar equations:

$$\sum F_x=0.$$

$$\sum F_y=0.$$

In three dimensions, a third equation is also required:

$$\sum F_z=0.$$

Motion and Displacement

Displacement is the change between final and initial position vectors:

$$\Delta\mathbf{r}=\mathbf{r}_f-\mathbf{r}_i.$$

Average velocity is displacement divided by elapsed time:

$$\mathbf{v}_{\mathrm{avg}}=\frac{\Delta\mathbf{r}}{\Delta t}.$$

Average acceleration is change in velocity divided by elapsed time:

$$\mathbf{a}_{\mathrm{avg}}=\frac{\Delta\mathbf{v}}{\Delta t}.$$

Summary

Vector operations provide a systematic way to analyze quantities with magnitude and direction. The most important operations include addition, subtraction, scalar multiplication, normalization, dot products, cross products, and projections.

Three core relationships are:

$$\mathbf{a}+\mathbf{b}=\langle a_x+b_x,\;a_y+b_y\rangle.$$

$$\mathbf{a}\cdot\mathbf{b}=\lVert\mathbf{a}\rVert\lVert\mathbf{b}\rVert\cos\theta.$$

$$\lVert\mathbf{a}\times\mathbf{b}\rVert=\lVert\mathbf{a}\rVert\lVert\mathbf{b}\rVert\sin\theta.$$