how to find the resultant vector

Resultant Vector Continued

The Resultant Vector

Contents

The Resultant Vector
Example 1: Head to Tail Addition in One-Dimension
- Question
- Step 1: Choose a scale and a reference direction
- Step 2: Choose one of the vectors and draw it as an arrow of the correct length in the correct direction
- Step 3: Take the next vector and draw it starting at the arrowhead of the previous vector
- Step 4: Take the next vector and draw it starting at the arrowhead of the previous vector
- Step 5: Take the next vector and draw it starting at the arrowhead of the previous vector
- Step 6: Draw the resultant, measure its length and find its direction

In previous lessons, you learnt about adding vectors together in one dimension. The same principle can be applied for vectors in two dimensions. The following examples show addition of vectors. Vectors that are parallel can be shifted to fall on a line. Vectors falling on the same line are called co-linear vectors. To add co-linear vectors we use the tail-to-head method you learnt earlier. In the figure below we remind you of the approach of adding co-linear vectors to get a resultant vector.

Adding co-linear vectors to get a resultant vector.

In the above figure the blue vectors are in the \(y\)-direction and the red vectors are in the \(x\)-direction. The two black vectors represent the resultants of the co-linear vectors graphically.

What we have done is implement the tail-to-head method of vector addition for the vertical set of vectors and the horizontal set of vectors.

Example 1: Head to Tail Addition in One-Dimension

Question

Use the graphical head-to-tail method to determine the resultant force on a rugby player if two players on his team are pushing him forwards with forces of \(\stackrel{\to }{{F}_{1}}\) = \(\text{600}\) \(\text{N}\) and \(\stackrel{\to }{{F}_{2}}\) = \(\text{900}\) \(\text{N}\) respectively and two players from the opposing team are pushing him backwards with forces of \(\stackrel{\to }{{F}_{3}}\) = \(\text{1 000}\) \(\text{N}\) and \(\stackrel{\to }{{F}_{4}}\) = \(\text{650}\) \(\text{N}\) respectively.

Step 1: Choose a scale and a reference direction

Let's choose a scale of \(\text{100}\) \(\text{N}\): \(\text{0.5}\) \(\text{cm}\) and for our diagram we will define the positive direction as to the right.

Step 2: Choose one of the vectors and draw it as an arrow of the correct length in the correct direction

We will start with drawing the vector \(\stackrel{\to }{{F}_{1}}\) = \(\text{600}\) \(\text{N}\) pointing in the positive direction.Using our scale of \(\text{0.5}\) \(\text{cm}\) : \(\text{100}\) \(\text{N}\), the length of the arrow must be \(\text{3}\) \(\text{cm}\) pointing to the right.

Step 3: Take the next vector and draw it starting at the arrowhead of the previous vector

The next vector is \(\stackrel{\to }{{F}_{2}}\) = \(\text{900}\) \(\text{N}\) in the same direction as \(\stackrel{\to }{{F}_{1}}\). Using the scale, the arrow should be \(\text{4.5}\) \(\text{cm}\) long and pointing to the right.

Step 4: Take the next vector and draw it starting at the arrowhead of the previous vector

The next vector is \(\stackrel{\to }{{F}_{3}}\) = \(\text{1 000}\) \(\text{N}\) in the opposite direction. Using the scale, this arrow should be \(\text{5}\) \(\text{cm}\) long and point to the left.

Note: We are working in one dimension so this arrow would be drawn on top of the first vectors to the left. This will get confusing so we'll draw it next to the actual line as well to show you what it looks like.

Step 5: Take the next vector and draw it starting at the arrowhead of the previous vector

The fourth vector is \(\stackrel{\to }{{F}_{4}}\) = \(\text{650}\) \(\text{N}\) in the opposite direction. Using the scale, this arrow must be \(\text{3.25}\) \(\text{cm}\) long and point to the left.

Step 6: Draw the resultant, measure its length and find its direction

We have now drawn all the force vectors that are being applied to the player. The resultant vector is the arrow which starts at the tail of the first vector and ends at the head of the last drawn vector.

The resultant vector measures \(\text{0.75}\) \(\text{cm}\) which, using our scale is equivalent to \(\text{150}\) \(\text{N}\) and points to the left (or the negative direction or the direction the opposing team members are pushing in).

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Tutorial Lessons

Scalars and Vectors

Graphical Representation of Vectors

Drawing Vectors

Properties of Vectors

Adding Vectors

Subtracting Vectors

Resultant Vector

Graphical Techniques of Vector Addition

Examples on Vector Addition

Algebraic Techniques of Vector Addition

Summary and Main Ideas

Resultant of Perpendicular Vectors

Compass Directions and Bearings

Resultant Vector Continued

Magnitude of the Resultant

Sketching Tail-to-Tail Method

Closed Vector Diagrams

Finding Magnitude With Pythagoras Theorem

Graphical Methods Continued

Example on Graphical Methods

Resultant Force on a Submarine

Algebraic Methods Continued

Components of Vectors

Vector Addition Using Components

Adding Vectors Using Components

Resultant Using Vectors

Summary and Main Ideas

Common Physical Quantities

Common Physical Quantities Continued

Force Board Experiment