AOS5 Topic 2: Scalar/Vector Resolutes

Scalar and vector resolutions are methods used to break down a given vector into its components along specified directions. These components are known as scalar and vector components, respectively.

Scalar Resolution

Scalar resolution involves breaking down a vector into its components along specific axes. These components are scalar quantities, meaning they have magnitude but no direction. Scalar resolution is often used when analyzing the effect of a force or motion along a particular direction.

For instance, if we have a vector \( \mathbf{v} \) in two-dimensional space, its scalar components along the \( x \) and \( y \) axes would be \( v_x \) and \( v_y \), respectively. These scalar components represent the magnitude of the vector along each axis.


Created with GeoGebra®, by Peter Sassman, Link


Vector Resolution

Vector resolution involves breaking down a vector into its components along specific axes while retaining their vector nature. These components are vectors themselves, preserving both magnitude and direction. Vector resolution is particularly useful when dealing with forces or velocities in different directions.

Continuing with the example of a vector \( \mathbf{v} \) in two-dimensional space, its vector components along the \( x \) and \( y \) axes would be \( \mathbf{v}_x \) and \( \mathbf{v}_y \), respectively. These vector components represent the direction and magnitude of the original vector along each axis.



Vector Resolution in a Plane

The resolution of a vector in a plane refers to the process of breaking down a given vector into its components along specific axes within that plane. In simpler terms, it involves expressing a vector as the sum of its parts in terms of direction and magnitude along chosen reference lines.

In a two-dimensional plane, such as the Cartesian coordinate system, vectors can be resolved into horizontal and vertical components. These components represent the contributions of the vector in the x-axis (horizontal) and y-axis (vertical) directions.

For example, if we have a vector 𝐯 in a two-dimensional plane, its resolution might involve determining how much of its magnitude is directed horizontally (along the x-axis) and how much is directed vertically (along the y-axis). This is typically achieved using trigonometric functions such as sine and cosine, in conjunction with the angle that the vector makes with the x-axis.

 

Below is the vector resolute of u in the direction of v

Created with GeoGebra®, by Tim Brzezinski, Link


Horizontal Component

The portion of a force that flows directly in a line parallel to the horizontal axis is known as the horizontal component in science.

Assuming you kick a football, the force of the kick may now be split into two parts: a vertical component that propels the football at an angle to the ground, and a horizontal component that pushes the football parallel to the ground.

Vertical Component

The portion or component of a vector that is perpendicular to a horizontal or level plane is known as the vertical component.



RESOLUTION OF A VECTOR IN A GIVEN BASIS

Consider two non-collinear vectors \( \vec{a} \) and \( \vec{b} \); as discussed earlier, these will form a basis of the plane in which they lie. Any vector \( \vec{r} \) in the plane of \( \vec{a} \) and \( \vec{b} \) can be expressed as a linear combination of \( \vec{a} \) and \( \vec{b} \):

The vector \( \overrightarrow{r} \) can be represented as \( \overrightarrow{r} = \overrightarrow{OA} \) + \overrightarrow{OB} =\lambda \overrightarrow{a} + \mu \overrightarrow{b} \) for some \( \lambda, \mu \mathbb{R} \).

The vectors \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \) are called the components of the vector \( \overrightarrow{r} \) along the basis formed by \( \overrightarrow{a} \) and \( \overrightarrow{b} \). This is also stated by saying that the vector \( \overrightarrow{r} \) when resolved along the basis formed by \( \overrightarrow{a} \) and \( \overrightarrow{b} \), gives the components \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \).

RECTANGULAR RESOLUTIONL

Let us select as the basis for a plane, a pair of unit vectors \( \hat{i} \) and \( \hat{j} \) perpendicular to each other.

Let \( \theta \) be the angle that \( \vec{r} \) makes with \( \hat{i} \):

\( OA = |\hat{r}| \cdot \cos \theta \)

\( OB = |\hat{j}| \cdot \sin \theta \)

Any vector \( \vec{r} \) in this basis can be written as:

\[ \vec{r} = \vec{OA} + \vec{OB} = (|\vec{r}| \cos \theta) \hat{i} + (|\vec{r}| \sin \theta) \hat{j} = x \hat{i} + y \hat{j} \]

where \( x \) and \( y \) are referred to as the \( x \) and \( y \) components of \( \vec{r} \).

For 3-D space, we select three unit vectors \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) each perpendicular to the other two.

In this case, any vector \( \vec{r} \) will have three corresponding components, generally denoted by \( x \), \( y \), and \( z \). We thus have

\[ \vec{r} = x \hat{i} + y \hat{j} + z \hat{k} \]

The basis \( (\hat{i}, \hat{j}) \) for the two-dimensional case and \( (\hat{i}, \hat{j}, \hat{k}) \) for the three-dimensional case are referred to as a rectangular basis and are extremely convenient to work with. Unless otherwise stated, we'll always be using a rectangular basis from now on.

Example 1

A position vector in two dimensions has magnitude \(5\) and its direction, measured anticlockwise from the x-axis, is \(150^\circ\). Express this vector in terms of \( \mathbf{i} \) and \( \mathbf{j} \).

Solution:

\(\text{Let } \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j}\).


The vector \( \mathbf{a} \) makes an angle of 150° with the \( x \)-axis and an angle of 60° with the \( y \)-axis. Therefore,


\(\cos 150^\circ = \frac{a_1}{|\mathbf{a}|}\)

and

\(\cos 60^\circ = \frac{a_2}{|\mathbf{a}|}\).



Since \( |\mathbf{a}| = 5 \), this gives \(a_1 = |\mathbf{a}| \cos 150^\circ = -\frac{5\sqrt{3}}{2}\)

and \(a_2 = |\mathbf{a}| \cos 60^\circ = \frac{5}{2}.\)

Hence, \(\mathbf{a} = -\frac{5\sqrt{3}}{2}\mathbf{i} + \frac{5}{2}\mathbf{j}\).

Exercise &&1&& (&&1&& Question)

Scalar resolution involves breaking down a vector into its components along specific axes. What does this process entail?

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Exercise &&2&& (&&1&& Question)

Sarah is pushing a box across the floor with a force of 50 Newtons at an angle of 30 degrees above the horizontal. Determine the horizontal and vertical components of the force applied by Sarah.

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