Vector Algebra

Exercise 1 : Distributivity of Dot and Cross

Prove that the dot and cross products are distributive, using definitions and diagrams:

  1. for three coplanar vectors
  2. in the general case

Exercise 2 : Cross Product Associativity?

Is the cross product associative? That is, does the following hold?

$$ (A \times B) \times C = A \times (B \times C) $$

Prove or provide a counterexample.

Exercise 3 : BAC-CAB Identity

Prove the vector identity:
$A \times (B \times C) = B,(A \cdot C) - C,(A \cdot B)$ (the BAC-CAB rule) by writing both sides in component form.

Exercise 4 : Jacobi Identity

Verify the Jacobi identity:

$$[A \times (B \times C)] + [B \times (C \times A)] + [C \times (A \times B)] = 0$$


Under what condition does $A \times (B \times C) = (A \times B) \times C$?

Exercise 5 : Cyclic Cross Products

Show that if $A + B + C = 0$, then $A \times B = B \times C = C \times A$.

Exercise 6 : Angle Between Body Diagonals

Find the angle between the body diagonals of a cube.

Exercise 7 : Unit Normal to a Plane

Find the components of the unit vector perpendicular to a plane defined by three given points or a diagram.

Exercise 8 : Rotation Matrices

  1. Prove that the 2D rotation matrix preserves dot products.
  2. What constraints must the elements $R_{ij}$ of a 3D rotation matrix satisfy to preserve vector lengths?

Exercise 9 : 120° Rotation About (1,1,1)

Find the transformation matrix for a $120^\circ$ clockwise rotation about the axis through $(1,1,1)$.

Exercise 10 : Coordinate Transformations

How do vector components transform under:

  1. translation: $\bar{x}=x,\ \bar{y}=y-a,\ \bar{z}=z$?
  2. inversion: $\bar{x}=-x,\ \bar{y}=-y,\ \bar{z}=-z$?
  3. How does a cross product transform under inversion? Is the cross product of two pseudovectors a vector or pseudovector? Give two examples of pseudovectors in mechanics.
  4. How does the scalar triple product transform under inversion?

Exercise 11 : Divergence as a Scalar (2D)

Show that in two dimensions, the divergence transforms as a scalar under rotations.

Exercise 12 : Gradient Transforms as a Vector

Show that the gradient of a scalar function transforms as a vector under rotations.

Exercise 13 : Compute Gradients

Compute the gradient of:

  1. $f(x,y,z) = x^2 + y^3 + z^4$
  2. $f(x,y,z) = x^2 y^3 z^4$
  3. $f(x,y,z) = e^x \sin y \ln z$

Exercise 14 : Hilltop and Slope

For the hill height \(h(x,y) = 10(2xy - 3x^2 - 4y^2 - 18x + 28y + 12)\):

  1. Locate the top of the hill.
  2. Find its height.
  3. Find the slope (in ft/mi) and direction of steepest ascent at \((1,1)\).

Exercise 15 : Gradients of Powers of r

For $r$ from $(x’,y’,z’)$ to $(x,y,z)$ with $r = |r|$:

  1. Show $\nabla(r^2) = 2r$
  2. Show $\nabla(1/r) = -\hat{r}/r^2$
  3. Find $\nabla(r^n)$ for integer $n$.

Exercise 16 : Divergence Calculations

Calculate the divergence of and divergence of:

  1. $v_a = x^2\hat{x} + 3xz^2\hat{y} - 2xz\hat{z}$
  2. $v_b = xy\hat{x} + 2yz\hat{y} + 3zx\hat{z}$
  3. $v_c = y^2\hat{x} + (2xy+z^2)\hat{y} + 2yz\hat{z}$

Exercise 17 : Divergence of Radial Field

Sketch $v = \hat{r}/r^2$ and compute its divergence. Explain any surprising result.

Exercise 18 : Tangential Field Curl

For a clockwise-tangential vector field around a circle in the $xy$-plane:

  1. Determine signs of $\partial v_x/\partial y$ and $\partial v_y/\partial x$
  2. Find the direction of $\nabla \times {v}$ and interpret geometrically.

Exercise 19 : Vector Calculus Identities

Verify these identities for arbitrary scalar field $\phi$ and vector field ${v}$:

  1. $\nabla \cdot (\phi {v}) = \phi(\nabla \cdot {v}) + {v} \cdot (\nabla \phi)$

  2. $\nabla \times (\phi {v}) = \phi(\nabla \times {v}) - {v} \times (\nabla \phi)$ Verify these identities for arbitrary scalar field $\phi$ and vector field $v$:

  3. $\nabla \cdot (\phi v) = \phi(\nabla \cdot v) + v \cdot (\nabla \phi)$

  4. $\nabla \times (\phi v) = \phi(\nabla \times v) - v \times (\nabla \phi)$

Exercise 20 : Electrostatic Field Candidates

Which of the following vector fields could represent an electrostatic field? (Electrostatic fields have zero curl.)

  1. $y\hat{x} + x\hat{y}$
  2. $y\hat{x} - x\hat{y}$
  3. $x\hat{x} + y\hat{y}$
  4. $x^2\hat{x} + 2xy\hat{y}$

Exercise 21 : Separation Vector

Find the separation vector from source point $(2,8,7)$ to field point $(4,6,8)$. Determine its magnitude and unit vector.

Exercise 22 : Point Charge Field Properties

If $E = kq\frac{r}{r^3}$ is the electric field of a point charge, verify that:

  1. $\nabla \cdot E = 0$ for $r \neq 0$
  2. $\nabla \times E = 0$ for $r \neq 0$ (What happens at $r=0$?)

Exercise 23 : Line Integrals and Conservativeness

For a vector function ${F} = (3x^2 + 2y)\hat{x} + (x - 4y^3)\hat{y}$:

  1. Compute the line integral from $(0,0)$ to $(1,1)$ along the path $y=x^2$
  2. Compute the same integral along the straight line $y=x$
  3. Is ${F}$ conservative? Explain.