MathIsimple
Lesson 3-3

Spatial Vectors – Dot/Cross and Applications

Use dot and cross products to compute angles, areas, and projections. Apply to geometric and physical problems.

Learning Objectives Back to Unit 3

Learning Objectives

  • Add, subtract, and scale vectors; interpret geometric meaning.
  • Compute dot product; derive angle and test orthogonality.
  • Compute cross product; determine area and orientation.
  • Find vector projections and components.
  • Apply vector tools to 3D geometry and basic physics contexts.

Core Concepts

Dot Product

a⃗⋅b⃗=axbx+ayby+azbz=∥a⃗∥ ∥b⃗∥cos⁡θ\vec{a}\cdot \vec{b}=a_xb_x+a_yb_y+a_zb_z=\|\vec{a}\|\,\|\vec{b}\|\cos\theta

Orthogonal ⇔ dot product 0.

Cross Product

a⃗×b⃗=∣i⃗j⃗k⃗axayazbxbybz∣\vec{a}\times \vec{b}=\begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \cr a_x & a_y & a_z \cr b_x & b_y & b_z \end{vmatrix}

Magnitude equals area of the parallelogram: ∥a⃗×b⃗∥=∥a⃗∥ ∥b⃗∥sin⁡θ\|\vec{a}\times \vec{b}\|=\|\vec{a}\|\,\|\vec{b}\|\sin\theta.

Projection

projb⃗a⃗=(a⃗⋅b⃗∥b⃗∥2)b⃗\text{proj}_{\vec{b}}\vec{a}=\left( \tfrac{\vec{a}\cdot \vec{b}}{\|\vec{b}\|^2} \right) \vec{b}

Scalar Triple Product

[a⃗,b⃗,c⃗]=a⃗⋅(b⃗×c⃗)[\vec{a},\vec{b},\vec{c}]=\vec{a}\cdot(\vec{b}\times \vec{c})

Absolute value equals volume of the parallelepiped.

Worked Examples

Example 1: Angle Between Vectors

For a⃗=(1,0,2),  b⃗=(2,1,1)\vec{a}=(1,0,2),\; \vec{b}=(2,1,1), find the angle.

cos⁡θ=a⃗⋅b⃗∥a⃗∥ ∥b⃗∥=1⋅2+0⋅1+2⋅112+02+22 22+12+12=45 6\cos\theta=\tfrac{\vec{a}\cdot \vec{b}}{\|\vec{a}\|\,\|\vec{b}\|}=\tfrac{1\cdot 2+0\cdot1+2\cdot1}{\sqrt{1^2+0^2+2^2}\,\sqrt{2^2+1^2+1^2}}=\tfrac{4}{\sqrt{5}\,\sqrt{6}}

Example 2: Parallelogram Area

Compute area spanned by a⃗=(1,0,2)\vec{a}=(1,0,2), b⃗=(2,1,1)\vec{b}=(2,1,1).

a⃗×b⃗=(−2,3,1),    ∥a⃗×b⃗∥=(−2)2+32+12=14\vec{a}\times \vec{b}=(-2,3,1),\; \; \|\vec{a}\times \vec{b}\|=\sqrt{(-2)^2+3^2+1^2}=\sqrt{14}

Example 3: Projection

Find projb⃗a⃗\text{proj}_{\vec{b}}\vec{a} for the same vectors.

projb⃗a⃗=(46)b⃗=23(2,1,1)=(4/3,2/3,2/3)\text{proj}_{\vec{b}}\vec{a}=\left(\tfrac{4}{6}\right)\vec{b}=\tfrac{2}{3}(2,1,1)=(4/3,2/3,2/3)

Example 4: Scalar Triple Product

For a⃗=(1,2,0),  b⃗=(0,1,3),  c⃗=(2,1,1)\vec{a}=(1,2,0),\; \vec{b}=(0,1,3),\; \vec{c}=(2,1,1), compute volume.

[a⃗,b⃗,c⃗]=det⁡[120013211]=1(1⋅1−3⋅1)−2(0⋅1−3⋅2)+0(0⋅1−1⋅2)=1+12=13[\vec{a},\vec{b},\vec{c}]=\det \begin{bmatrix} 1 & 2 & 0 \cr 0 & 1 & 3 \cr 2 & 1 & 1 \end{bmatrix}=1(1\cdot1-3\cdot1)-2(0\cdot1-3\cdot2)+0(0\cdot1-1\cdot2)=1+12=13

Example 5: Orthogonality Test

Show that (1,−2,1)(1,-2,1) and (2,1,0)(2,1,0) are perpendicular.

(1)(2)+(−2)(1)+(1)(0)=0(1)(2)+(-2)(1)+(1)(0)=0

Practice Problems

Problem 1

For u⃗=(3,−1,2),  v⃗=(−1,4,0)\vec{u}=(3,-1,2),\; \vec{v}=(-1,4,0), compute u⃗⋅v⃗\vec{u}\cdot \vec{v} and angle.

Show Solution

Dot = −3−4+0=−7-3-4+0=-7; use norms to get cos⁡θ=−714 17\cos\theta=\tfrac{-7}{\sqrt{14}\,\sqrt{17}}.

Problem 2

Find area of triangle with sides a⃗=(1,2,3),  b⃗=(0,1,2)\vec{a}=(1,2,3),\; \vec{b}=(0,1,2).

Show Solution

Triangle area = 12∥a⃗×b⃗∥\tfrac{1}{2}\|\vec{a}\times \vec{b}\|.

Problem 3

Compute projection of (2,3,4)(2,3,4) on (1,0,1)(1,0,1).

Show Solution

(2⋅1+3⋅0+4⋅112+02+12)(1,0,1)=(3/2)(1,0,1)\left( \tfrac{2\cdot1+3\cdot0+4\cdot1}{1^2+0^2+1^2} \right)(1,0,1)=(3/2)(1,0,1).

Problem 4

Compute [a⃗,b⃗,c⃗][\vec{a},\vec{b},\vec{c}] for a⃗=(1,0,0),  b⃗=(0,1,0),  c⃗=(0,0,1)\vec{a}=(1,0,0),\; \vec{b}=(0,1,0),\; \vec{c}=(0,0,1).

Show Solution

=1 (unit volume).

Problem 5

Show (1,1,1)(1,1,1) is orthogonal to (1,−1,0)(1,-1,0) + (0,1,−1)(0,1,-1).

Show Solution

Sum = (1,0,-1). Dot with (1,1,1) is 0.

Key Takeaways

  • Dot product links to angles and orthogonality; cross product to areas and perpendicular vectors.
  • Projections decompose vectors along directions useful for physics components.
  • Scalar triple product gives signed volume and orientation.
  • Vector tools unify geometry computations elegantly in 3D.

Extensions: Vector Identities & Applications

Vector Identities

a⃗×(b⃗×c⃗)=(a⃗⋅c⃗) b⃗−(a⃗⋅b⃗) c⃗\vec{a}\times(\vec{b}\times \vec{c})= (\vec{a}\cdot \vec{c})\,\vec{b} - (\vec{a}\cdot \vec{b})\,\vec{c}
(a⃗×b⃗)⋅(c⃗×d⃗)=(a⃗⋅c⃗)(b⃗⋅d⃗)−(a⃗⋅d⃗)(b⃗⋅c⃗)(\vec{a}\times \vec{b})\cdot(\vec{c}\times \vec{d})= (\vec{a}\cdot \vec{c})(\vec{b}\cdot \vec{d})-(\vec{a}\cdot \vec{d})(\vec{b}\cdot \vec{c})

Physics: Torque & Work

Work uses dot product; torque uses cross product.

W=F⃗⋅s⃗W=\vec{F}\cdot \vec{s}
τ⃗=r⃗×F⃗\vec{\tau}=\vec{r}\times \vec{F}

Advanced Applications

Magnetic Force

The Lorentz force on a charged particle in magnetic field.

F⃗=q(v⃗×B⃗)\vec{F}=q(\vec{v}\times \vec{B})

Direction follows right-hand rule.

Angular Momentum

Position vector crossed with momentum vector.

L⃗=r⃗×p⃗=r⃗×(mv⃗)\vec{L}=\vec{r}\times \vec{p}=\vec{r}\times (m\vec{v})

Electric Field from Current

Biot-Savart law relates current to magnetic field.

dB⃗=μ0I4πdl⃗×r⃗r3d\vec{B}=\frac{\mu_0 I}{4\pi}\frac{d\vec{l}\times \vec{r}}{r^3}

Vector Fields

Divergence and curl operations on vector fields.

∇⋅F⃗=∂Fx∂x+∂Fy∂y+∂Fz∂z\nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}

Comprehensive Problem Bank

Problem Set A: Basic Operations

  1. Compute (2,1,−3)⋅(−1,4,2)(2,1,-3)\cdot(-1,4,2) and explain significance of result.
  2. Find (1,2,0)×(3,−1,4)(1,2,0)\times(3,-1,4) and verify orthogonality.
  3. Normalize (3,4,12)(3,4,12) and check unit length.
  4. Project (5,2,1)(5,2,1) onto (1,1,1)(1,1,1).

Problem Set B: Geometric Applications

  1. Find area of triangle with vertices A(1,0,0), B(0,1,0), C(0,0,1).
  2. Compute volume of tetrahedron with fourth vertex at origin.
  3. Find unit normal to plane containing three given points.
  4. Determine if four points are coplanar using scalar triple product.

Problem Set C: Physics Contexts

  1. Force F⃗=(3,4,0)\vec{F}=(3,4,0) N applied for displacement s⃗=(2,1,3)\vec{s}=(2,1,3) m. Find work done.
  2. Torque about origin for force at position. Compute magnitude and direction.
  3. Find resultant of three forces and equilibrium conditions.
  4. Angular velocity ω⃗\vec{\omega} and radius r⃗\vec{r}. Find linear velocity.

Problem Set D: Advanced Topics

  1. Prove BAC-CAB identity using component expansion.
  2. Show that a⃗×(b⃗×c⃗)+b⃗×(c⃗×a⃗)+c⃗×(a⃗×b⃗)=0⃗\vec{a}\times(\vec{b}\times \vec{c}) + \vec{b}\times(\vec{c}\times \vec{a}) + \vec{c}\times(\vec{a}\times \vec{b}) = \vec{0}.
  3. Find all vectors orthogonal to both (1,2,3)(1,2,3) and (4,5,6)(4,5,6).
  4. Derive formula for distance between two skew lines using vectors.

Extended FAQ & Troubleshooting

Common Misconceptions

Q: Is vector multiplication commutative?

A: Dot product yes, cross product no: a⃗×b⃗=−b⃗×a⃗\vec{a}\times\vec{b} = -\vec{b}\times\vec{a}.

Q: Can cross product magnitude exceed factor magnitudes?

A: No, ∥a⃗×b⃗∥≤∥a⃗∥ ∥b⃗∥\|\vec{a}\times\vec{b}\| \leq \|\vec{a}\|\,\|\vec{b}\| with equality when perpendicular.

Computational Tips

Q: How to avoid determinant errors in cross products?

A: Use systematic expansion: first component uses ∣ayazbybz∣\begin{vmatrix} a_y & a_z \\ b_y & b_z \end{vmatrix} with positive sign.

Q: Quick orthogonality check?

A: Compute dot product; zero result confirms perpendicularity.

Extended Practice Sets

Set 1: Triple Products

  1. Compute scalar and vector triple products.
  2. Interpret signed volume.
  3. Check orientation changes.

Set 2: Triangle Area & Normal

  1. Use cross product to find area.
  2. Normalize to get unit normal.
  3. Verify orientation via right-hand rule.

Set 3: Force Resolution

  1. Resolve along/against a plane.
  2. Compute work via dot product.
  3. Torque via cross product.

Set 4: BAC–CAB Identity

  1. Simplify a⃗×(b⃗×c⃗)\vec{a}\times(\vec{b}\times \vec{c}).
  2. Plug into a mechanics example.
  3. Discuss computational efficiency.
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