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3. Vectors

3.1 Introduction

A quantity that has both a magnitude and direction is called a vector. It is common to denote a vector symbolically by an arrow on top such as \(\vec{v}\).

In this course, we will study several vector quantities. We will start with displacement (\(\vec{s}\)), velocity (\(\vec{v}\)), and acceleration (\(\vec{a}\)) vectors in the study of motion. Then, we will go on to study other important vector quantities such as force (\(\vec{F}\)), momentum (\(\vec{p}\)) and their rotational counterparts torque (\(\vec{\tau}\)) and angular momentum (\(\vec{\ell}\)).

3.2 Unit vector

A vector of unit magnitude is called a unit vector and is denoted by a hat symbol rather than an arrow symbol. For example:

  • \(\hat{i}\) is a unit vector pointing in the positive-\(x\) direction,
  • \(\hat{j}\) is a unit vector pointing in the positive-\(y\) direction, and
  • \(\hat{k}\) is a unit vector pointing in the positive-\(z\) direction.

Product of a scalar and a vector results in a new vector. The direction of the new vector is the same if the scalar is positive, whereas the direction of the new vector is opposite if the scalar is negative. The magnitude of the new vector is scaled by the magnitude of the scalar.

Use the graphical slider control below to see how the multiplication of a unit vector by a real number rescales the unit vector.

\[ \vec{a} = a\ \hat{i} \]

A vector in 2D can be represented by two numbers \(a_x\) and \(a_y\) that scale the unit vectors in \(x\) and \(y\) directions:

\[ \vec{a} = a_x \hat{i} + a_y \hat{j} \]

Vector scaling in 2D

Use the two sliders to scale the x and y components

\(\vec{a}=\) 1.0 \(\hat{i}\ \)+ 1.0 \(\hat{j}\)

3.3 Mathematical representation of a vector in 3D

A vector of any length and direction in three dimensions can be represented by three numbers \(a_x, a_y,\) and \(a_z\) scaling the unit vectors in each of the \(x, y\) and \(z\) directions:

\[ \vec{a} = a_x\ \hat{i} + a_y\ \hat{j} + a_z\ \hat{k} \]
\[ \vec{a} = \begin{pmatrix} a_x \\ a_y \\ a_z \end{pmatrix} \]

Graphical

Above, you see a vector (solid red) and its components (dashed and dotted lines). Click on the figure and use your mouse buttons to rotate and/or zoom the 3D figure.

3.4 Vector Magnitude

The magnitude of the vector \(\vec{a}\) is given by:

\[ a = |\vec{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2} \]

Proof: Using Pythagorean theorem, the length of the vector \(\vec{a}\) can be found as: \(a = OA = \sqrt{ OB^2 + a_z^2}\), where \(OB^2 = a_x^2 + a_y^2\).

If \(\vec{a} = 2 \hat{i} - 3 \hat{j} + \hat{k}\), find \(|\vec{a}|\).

Solution: \(|\vec{a}| = \sqrt{2^2 + (-3)^2 + 1^2} = \sqrt{14}\).

If \(\vec{b} = 2 \hat{j} + \hat{k}\), find \(|\vec{b}|\).

Solution: \(|\vec{b}| = \sqrt{0^2 + 2^2 + 1^2} = \sqrt{5}\).

3.5 Vector Operations

If \(\vec{a} = a_x \hat{i} + a_y \hat{j} + a_z \hat{k}\) and \(\vec{b} = b_x \hat{i} + b_y \hat{j} + b_z \hat{k}\), then the sum of two vectors \(\vec{c} = \vec{a}+\vec{b}\) is a new vector whose \(x\), \(y\) and \(z\) components can be found by adding the respective components of the two vectors:

\[c_x = a_x + b_x, \quad c_y = a_y + b_y,\ {\rm and} \quad c_z = a_z+b_z.\]

If \(\vec{a} = a_x \hat{i} + a_y \hat{j} + a_z \hat{k}\) is multiplied by a scalar \(m\), then the result \(m \vec{a}\) is a new vector. The components of the new vector \(\vec{d} = m \vec{a}\) can be found by scaling the respective components of \(\vec{a}\) by \(m\):

\[d = m \vec{a} = (m\ a_x) \hat{i} + (m\ a_y) \hat{j} + (m\ a_z) \hat{k}\]

If \(\vec{a} = 2 \hat{i} -3 \hat{j}\) and \(\vec{b}= 2 \hat{j} + \hat{k}\), find \(2\vec{a} + \vec{b}\).

\[\begin{align}2\vec{a} + \vec{b} &= 2(2 \hat{i} - 3 \hat{j}) + (2 \hat{j}+\hat{k})\\ &= 4 \hat{i} - 6 \hat{j} + 2 \hat{j} + \hat{k} \\ &= 4 \hat{i} - 4\hat{j} + \hat{k} \end{align}\]

3.6 Vector Products

Dot product is a product of two vectors resulting in a scalar.

The geometric definition of the dot product of two vectors \(\vec{a}\) and \(\vec{b}\) is as follows:

\[ \vec{a}\cdot\vec{b} = |\vec{a}| |\vec{b}| \cos \phi \]

where \(\phi\) is the angle between \(\vec{a}\) and \(\vec{b}\).

Using this geometric definition, it is possible to work out the dot product between the three unit vectors:

  • \(\hat{i}\cdot \hat{i} = |\hat{i}||\hat{i}| \cos 0^\circ = 1\)
  • \(\hat{j}\cdot \hat{j} = |\hat{j}||\hat{j}| \cos 0^\circ = 1\)
  • \(\hat{k}\cdot \hat{k} = |\hat{k}||\hat{k}| \cos 0^\circ = 1\)
  • \(\hat{i}\cdot \hat{j} = |\hat{i}||\hat{j}| \cos 90^\circ = 0\)
  • \(\hat{j}\cdot \hat{k} = |\hat{j}||\hat{k}| \cos 90^\circ = 0\)

In general, if we have two vectors

\[ \vec{a} = a_x \hat{i} + a_y \hat{j} + a_z \hat{k}, \qquad {\rm and} \qquad \vec{b} = b_x \hat{i} + b_y \hat{j} + b_z \hat{k} \]

then,

\[ \begin{align} \vec{a}\cdot \vec{b} &= (a_x \hat{i} + a_y \hat{j} + a_z \hat{k})\cdot (b_x \hat{i} + b_y \hat{j} + b_z \hat{k}) \\ &= a_x b_x \ (\hat{i}\cdot\hat{i}) \\ &= a_x b_x + a_y b_y + a_z b_z \end{align} \]

Cross product of two vectors produces a new vector. The magnitude of the new vector is given by the product of the magnitudes of the two vectors times the sine of the angle between them

\[ |\vec{a}\times \vec{b}| = |\vec{a}||\vec{b}| \sin \phi \]

where \(\phi\) is the angle between \(\vec{a}\) and \(\vec{b}\).

The direction of the new vector given by the cross product \(\vec{a}\times \vec{b}\) can be found using the right hand rule: lay the fingers of your right hand in the direction of the first vector \(\vec{a}\), curl them towards the second vector \(\vec{b}\), then the thumb points in the direction of the cross product \(\vec{a}\times \vec{b}\). The direction of \(\vec{a}\times \vec{b}\) is perpendicular to both \(\vec{a}\) and \(\vec{b}\).

Using the above definition, we can obtain the following results for the cross product of unit vectors in cartesian co-ordinate system:

  • \(\hat{i}\times \hat{j} = \hat{k}\)
  • \(\hat{j}\times \hat{k} = \hat{i}\)
  • \(\hat{k}\times \hat{i} = \hat{j}\)
  • \(\hat{i}\times \hat{k} = -\hat{j}\)
  • \(\hat{i}\times \hat{i} = 0\)

If \(\vec{a} = 2\hat{i}+3\hat{j}\) and \(\vec{b}=4\hat{j}-3\hat{k}\), find \(\vec{a}\cdot \vec{b}\) and \(\vec{a} \times \vec{b}\)

\[\begin{align}\vec{a}\cdot \vec{b} = (2 \hat{i}+3 \hat{j})\cdot (4 \hat{j}- 3 \hat{k}) &= 8 (\hat{i}\cdot \hat{j})-6(\hat{i}\cdot \hat{k})+12(\hat{j}\cdot \hat{j})-9 (\hat{j}\cdot \hat{k}) \\ &= 8\times 0 -6 \times 0 + 12 \times 1 - 9\times 0 \\ &= 12 \end{align}\]
\[\begin{align}\vec{a}\times \vec{b} = (2 \hat{i}+3 \hat{j})\times (4 \hat{j} - 3 \hat{k}) &= 8(\hat{i}\times \hat{j}) - 6 (\hat{i}\times \hat{k}) + 12(\hat{j}\times \hat{j}) - 9(\hat{j}\times \hat{k}) \\ &= 8 \hat{k}+6\hat{j}+0-9\hat{i} \\ &= -9 \hat{i}+6\hat{j}+8\hat{k} \end{align}\]