Word Lesson: Vectors Right Triangles - Components and Resultants

• Know the relationships among rectangular vector coordinates, polar coordinates, magnitude, and direction
• Apply vector diagrams to find components from given information
• Find an angle given a right triangle ratio of sides
• Use the Pythagorean Theorem

A typical problem involving vectors and right triangles gives us information about a vector and/or its components. We are asked to find information about unmeasured components and angles. Note that we will represent vectors either by using bold type v or by using the vector notation .

First we make a diagram. In this case, the plane’s path makes a 35° angle with the ground. We call this vector p.

 

 

The hypotenuse of the triangle is given a length of 200 mph which is the magnitude of the vector p describing the plane’s path. The vector representing climbing vertically away from the ground is c and the vector representing the ground speed is g. The right angle is formed between vectors c and g which are components of p.

 

 

An easy way to identify which vectors are the components and which vector is the resultant (or sum of those two components) is to trace their "head-to-tail" relationship.
When adding vectors, we start by placing the head of the first vector (g) at the original of a co-ordinate system.
At its tail we place the origin of a second co-ordinate system and the head of the second vector (c).
The resultant (p) is the vector that begins at the head of vector 1 (g) and ends at the tail of vector 2 (c) as shown in the diagram to the right.

 

In the diagram shown below the known magnitude (||p|| = 200 mph) and angle (q = 35º) are labeled as well as the unknown vectors corresponding to the situation: the magnitude of the climbing speed of the plane by c, and the magnitude of the ground speed of the plane by g.

 

 

Using the right triangle ratios sine and cosine we have:

 

 

Thus the climbing and ground speeds are approximately 114.7 mph and 163.8 mph respectively and are the vertical and horizontal components of the resultant vector p.

Directions and/or Common Information:

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A force of 100 pounds is required to pull a 400 pound block up a ramp at a constant speed. What angle does the ramp make with respect to the ground?

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A plane is traveling at 400 mph bearing 20° (bearings are measured from North in a clockwise manner). How fast is the plane traveling North? How fast is the plane traveling East?

 

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