Subtracting Geometric Vectors

Subtracting vectors results in a new Resultant Vector. Geometric and Algebraic Vectors can be subtracted using different techniques.

Geometric vectors can be subtracted by positioning the vectors Head to Tail or Tail to Tail. Remember, you can move a vector around in space as long as it maintains its magnitude and direction. Subtracting vectors is the same as adding the Opposite Vector.

Subtracting Geometric Vectors Head to Tail

To subtract vectors head to tail, add the Opposite vector. Move the Opposite vector so that it's tail starts at the head of the positive vector. The Resultant Vector starts at the tail of the first vector and ends at the head of the last one.

Below are the steps to subtract vectors \(\vec{x}\) and \(\vec{w}\) head to tail. This method is referred to the Parallelogram Law of Vector Addition but we are adding the Opposite vector.

Vectors x and -w arranged head to tail with resultant vector.

The figure below shows how to add and subtract \(\vec{x}\) and \(\vec{w}\) :

Figure of Vector x arranged head to tail with Vectors w and -w, creating resultants of x+w and x-w.

Subtracting Geometric Vectors involves solving triangles using trigonometry equations. Below is a summary:

Right Triangles
Pythagorean Theorem	\(c^2 = a^2 + b^2 \)
SOH CAH TOA	\(\sin(\theta) = \cfrac{\text{opposite}}{\text{hypotenuse}} = \cfrac{y}{r}\) \(\cos(\theta) = \cfrac{\text{adjacent}}{\text{hypotenuse}} = \cfrac{x}{r} \) \(\tan(\theta) = \cfrac{\text{opposite}}{\text{adjacent}} = \cfrac{y}{x}\)

Other Triangles
Cosine Law	\(c^2 = a^2 + b^2 - 2 a b \cos(\theta)\)
Sin Law	\(\cfrac{\sin(A)}{a} = \cfrac{\sin(B)}{b} = \cfrac{\sin(C)}{c} \)

Review these lessons:

Try these questions:

A plane travels \(200\; \left[\cfrac{\text{km}}{\text{hr}}\right] N 20^\circ E \) when the wind speed is \(90 \left[\cfrac{\text{km}}{\text{hr}}\right] E \). What is the plane's velocity when there is no wind?

Show Answer

The plane's overall velocity with respect to the ground is \(\mathbf{v} = \mathbf{p} + \mathbf{w} \) where \(\mathbf{p}\) is the plane's speed and \( \mathbf{w}\) is the wind speed. Therefore, \( \mathbf{p} = \mathbf{v} - \mathbf{w}\).

Subtract \(90 [\frac{km}{hr}] E \) by adding a vector \( 90 [\frac{km}{hr}] W \):

Velocity vectors arranged head to tail with place speed as the resultant vector.

Use Cosine Law to calculate the magnitude:

\(|p|^2 = 90^2 + 200^2 - (2)(90)(200)\cos(70^\circ)\)

\(|p| = \sqrt{35,787} \approx 189 \left[\cfrac{\text{km}}{\text{hr}}\right]\)

Use Sine Law to find the angle:

\(\cfrac{\sin(\theta)}{90} = \cfrac{\sin(70^\circ)}{189} \)

\(\theta = \sin^{-1}\left(\sin(70^\circ)\cdot \cfrac{90}{189}\right) \approx 27^\circ\)

Therefore, we can determine that the plane's velocity is \(\boldsymbol{189 \; N 7^\circ W}\).

Subtracting Geometric Vectors Tail to Tail

To subtract vectors Tail to Tail, move the negative vector so that it's tail starts at the tail of the positive vector. The Resultant Vector starts at the head of the negative vector and ends at the head of the positive vector.

Vectors w and x arranged tail to tail, with resultant vector of x-w.

The figure below shows how to add and subtract \(\vec{x}\) and \(\vec{w}\) :

Vectors x and w arranged tail to tail in couple of different ways, creating 2 resultant vectors.

In parallelogram ABCDE, what is \(\vec{AB} - \vec{AD} \)?

Show Answer

We can use the addition collapsing in conjuntion with opposite vectors:

\(=-\vec{BA} - \vec{AD} \)

\(=-(\vec{BA} + \vec{AD}) \)

\(=-(\vec{B\require{cancel}\cancel{A}} + \vec{\cancel{A}D}) \)

\(=-\vec{BD} \)

\(=\vec{DB} \)

This makes sense since the Resultant Vector starts at the head of the negative vector (\(\vec{D}\)) and ends at the head of the positive vector (\(\vec{B}\)).