Scalar and Vector Products

Exercise 1 : Scalar Products

Calculate the scalar products of the vectors $\mathbf{a}$ and $\mathbf{b}$ given below:

  1. $a = 3$, $b = 2$, $\alpha = \pi/3$
  2. $a = 2$, $b = 5$, $\alpha = 0$
  3. $a = 1$, $b = 4$, $\alpha = \pi/4$
  4. $a = 2.5$, $b = 3$, $\alpha = 120^\circ$

Exercise 2 : Angle from Scalar Product

Considering the scalar products, what can you say about the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$?

  1. $\mathbf{a} \cdot \mathbf{b} = 0$
  2. $\mathbf{a} \cdot \mathbf{b} = ab$
  3. $\mathbf{a} \cdot \mathbf{b} = \dfrac{ab}{2}$
  4. $\mathbf{a} \cdot \mathbf{b} < 0$

Exercise 3 : Component-wise Scalar Products

Calculate the scalar product of the following vectors:

  1. $\mathbf{a} = (3, -1, 4)$, $\mathbf{b} = (-1, 2, 5)$
  2. $\mathbf{a} = (3/2, 1/4, -1/3)$, $\mathbf{b} = (1/6, -2, 3)$
  3. $\mathbf{a} = (-1/4, 2, -1)$, $\mathbf{b} = (1, 1/2, 5/3)$
  4. $\mathbf{a} = (1, -6, 1)$, $\mathbf{b} = (-1, -1, -1)$

Exercise 4 : Perpendicular Vectors (Orthogonality)

Which of the following vectors $\mathbf{a}$ and $\mathbf{b}$ are perpendicular?

  1. $\mathbf{a} = (0, -1, 1)$, $\mathbf{b} = (1, 0, 0)$
  2. $\mathbf{a} = (2, -3, 1)$, $\mathbf{b} = (-1, 4, 2)$
  3. $\mathbf{a} = (-1, 2, -5)$, $\mathbf{b} = (-8, 1, 2)$
  4. $\mathbf{a} = (4, -3, 1)$, $\mathbf{b} = (-1, -2, -2)$
  5. $\mathbf{a} = (2, 1, -2)$, $\mathbf{b} = (-1, 3, -2)$
  6. $\mathbf{a} = (4, 2, 2)$, $\mathbf{b} = (1, -4, 2)$

Exercise 5 : Angle Between Vectors

Calculate the angle between the two vectors $\mathbf{a}$ and $\mathbf{b}$:

  1. $\mathbf{a} = (1, -1, 1)$, $\mathbf{b} = (-1, 1, -1)$
  2. $\mathbf{a} = (-2, 2, -1)$, $\mathbf{b} = (0, 3, 0)$

Exercise 6 : Work Done by a Force

A force $\mathbf{F} = (0,\text{N}, 5,\text{N})$ is applied to a body and moves it through a distance $\mathbf{s}$. Calculate the work done by the force.

  1. $\mathbf{s}_1 = (3,\text{m}, 3,\text{m})$
  2. $\mathbf{s}_2 = (2,\text{m}, 1,\text{m})$
  3. $\mathbf{s}_3 = (2,\text{m}, 0,\text{m})$

Exercise 7 : Direction of the Cross Product

Indicate in figures 2.22 and 2.23 the direction of the vector $\mathbf{c}$ if $\mathbf{c} = \mathbf{a} \times \mathbf{b}$:

  1. when $\mathbf{a}$ and $\mathbf{b}$ lie in the $x$-$y$ plane
  2. when $\mathbf{a}$ and $\mathbf{b}$ lie in the $y$-$z$ plane

Exercise 8 : Magnitude of the Vector Product

Calculate the magnitude of the vector product of the following vectors:

  1. $a = 2$, $b = 3$, $\alpha = 60^\circ$
  2. $a = 1/2$, $b = 4$, $\alpha = 0^\circ$
  3. $a = 8$, $b = 3/4$, $\alpha = 90^\circ$

Exercise 9 : Scalar Triple Product and Volume

Use the scalar triple product to compute the signed volume of the parallelepiped defined by the three vectors.

  1. $\mathbf{a}=(1,0,1)$, $\mathbf{b}=(0,2,1)$, $\mathbf{c}=(1,1,0)$
  2. $\mathbf{a}=(2,-1,0)$, $\mathbf{b}=(1,1,1)$, $\mathbf{c}=(0,1,2)$
  3. $\mathbf{a}=(1,2,3)$, $\mathbf{b}=(-1,0,1)$, $\mathbf{c}=(2,1,-1)$

Exercise 10 : Projection and Components

Compute the projection of $\mathbf{a}$ onto $\mathbf{b}$ and the perpendicular component $\mathbf{a}\perp = \mathbf{a}-\text{proj}{\mathbf{b}}(\mathbf{a})$.

  1. $\mathbf{a}=(3,4,0)$, $\mathbf{b}=(1,0,0)$
  2. $\mathbf{a}=(1,2,2)$, $\mathbf{b}=(2,1,0)$
  3. $\mathbf{a}=(-1,1,3)$, $\mathbf{b}=(1,1,1)$

Exercise 11 : Torque (Cross Product Application)

Calculate the torque $\boldsymbol{\tau}=\mathbf{r}\times\mathbf{F}$ and its magnitude for the given position and force vectors.

  1. $\mathbf{r}=(0.5,0,0)$ m, $\mathbf{F}=(0,10,0)$ N
  2. $\mathbf{r}=(1,1,0)$ m, $\mathbf{F}=(0,5,5)$ N
  3. $\mathbf{r}=(0,2,-1)$ m, $\mathbf{F}=(3,0,4)$ N

Exercise 12 : Area of a Parallelogram from Cross Product

Use $|\mathbf{a}\times\mathbf{b}|$ to compute the area of the parallelogram spanned by $\mathbf{a}$ and $\mathbf{b}$.

  1. $\mathbf{a}=(2,0,0)$, $\mathbf{b}=(0,3,0)$
  2. $\mathbf{a}=(1,2,2)$, $\mathbf{b}=(2,-1,1)$
  3. $\mathbf{a}=(1,1,0)$, $\mathbf{b}=(1,0,1)$

Exercise 13 : Gram–Schmidt Orthonormalization

Apply Gram–Schmidt to orthonormalize the given set of vectors.

  1. ${(1,1,0),(1,0,1)}$
  2. ${(1,2,2),(2,1,1)}$

Exercise 14 : True/False and Properties

Decide whether the statements are true or false; justify briefly.

  1. $(\mathbf{a}\times\mathbf{b})\cdot\mathbf{c}=(\mathbf{b}\times\mathbf{c})\cdot\mathbf{a}$ (cyclic permutation).
  2. $\mathbf{a}\times\mathbf{b}=\mathbf{b}\times\mathbf{a}$.
  3. $(k\mathbf{a})\times\mathbf{b}=k(\mathbf{a}\times\mathbf{b})$ for scalar $k$.
  4. $\mathbf{a}\cdot(\mathbf{b}+\mathbf{c})=\mathbf{a}\cdot\mathbf{b}+\mathbf{a}\cdot\mathbf{c}$.