Vector+components

=**Vector components**= include page="foo"

A **vector** is any quantity that can be described as a magnitude (length, amount, strength, etc.) and direction. For example, if a plane flies five miles north, its displacement is a vector with magnitude five miles and direction north.

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**Conceptual Introduction**
If you have a diagonal vector, you can split it into two vectors which add up to the original vector. These two vectors are called **components.** Any vector in two dimensions can be split into its two (x and y) components. This is a useful technique, because it allows you to work with each component separately. For example, if you have a diagonal velocity vector, gravity only works vertically, so you need to split it into component form in order to do any calculations with gravity. In general, **whenever you see a diagonal vector, you'll probably need to split it into its components.**

**Using Trigonometry to Find Vector Components**
You can use this method to find the components of a vector:
 * 1) Sketch a diagram of the vector, labeling its magnitude and direction. (for the direction, you'll need to be able to determine the angle from the vector to the horizontal or vertical.)
 * 2) Draw a rectangle such that the vector is a diagonal.
 * 3) Draw (with arrows) and label the components of the vector.
 * 4) Use the sine function to find the length of the opposite side, and use the cosine function to find the length of the adjacent side.

**Vector Decomposition**
Decomposing is the process of finding a vector’s components. Vector decomposition is useful when you have to add two vectors that are not parallel or perpendicular. In these cases, you need to resolve one vector into components that run parallel and perpendicular to the other vector. In the example, illustrated bellow, vector //**A**//, with a magnitude of //A// and a direction. Since Ax ( adjacent ), Ay (opposite) , and //**A**// (hypotenuse) form a right triangle. With these factors trigonometry can be used to solve this problem by applying the definitions of cosine and sine.



Trig Review
math \sin{\theta} = \dfrac{opposite}{hypotenuse}

\cos{\theta} = \dfrac{adjacent}{hypotenuse} math

**Example 1: Using Components to Add Vectors**
//Two ropes are tied to a box. One rope pulls east with a force of 2.0N. The second rope pulls with a force of 4.0N at an angle of 35º from the vertical, as shown in the diagram. Assuming that the surface of the box is fricionless, what is the total force acting on the box?//



**Example 2: A Diagonal Force**
//A dog exerts 80 N on his owner, 32 degrees northwest. If this dog were replaced with two dogs, one pulling north and one pulling west, how much force would each need to exert in order to have the same effect on the owner?//

**Magnitude and Direction with Components**
Pythagorean relationships and triangle trigonometry can be used to find both the magnitude and direction of a vector if the components are known. This is also known as the **polar form** of a vector.

**Example 3: Findin the Magnitude and Direction of a Vector**
//Find the magnitude and direction of vector A if Ax is equal to 13 and Ay is equal to 17.//

**Example 4: Finding the Magnitude and Direction of a Vector II**
//Find the magnitude and direction of vector **A** if Ax is equal to 6 and Ay is equal to 8.//

**Multiple Vector Addition**
To find the sum of vectors the components must first be known and be in polar form. Adding them together will give you the **resultant**. The **resultant** is a vector equal to two or more vectors.

**Example 5: Addition of Two Vectors**
//A car travels 5km to the east on road A and then turns on to road B and travels 8km northeast on road B. Find the direction and magnitude of the car if angle A equals 97° and angle B equals 30°.//

**Example 6: Addition of Three Vectors**
//In art class you build a structure made of three pipe cleaners. Pipe cleaner A is 9cm long with an angle of 70°, pipe cleaner B is 6cm long with an angle of 105° and pipe cleaner C is 4cm long with an angle of 20°. To make the structure stand up on its own you need to find the length of pipe cleaner, **R**, and the angle needed to connect the initial point of pipe cleaner A to the terminal point of pipe cleaner C.//

**Example 1**
Step 1, resolve the force on the second rope into its north and west components.



//4.0 cos30⁰= 4.0 × 0.866 = 3.464 north// //4.0 sin30⁰ = 4.0 × 0.5 = 2.0 west//

Step 2, since the east component is also 2.0 N, the east and west components end up cancelling each other out. The resulting force is directed north, with a force of about 3.4 N. This answer can be justified by using the parallelogram method. In using the parallelogram method you will find that the opposite corner of the parallelogram is directly above the corner made by the tails of the other two vectors. This can be seen in the diagram bellow.



**Example 2**
For step 1, sketch a free-body diagram: Step 2: Step 3: Step 4: math \sin{32\textdegree} = \dfrac{F_n}{80}

F_n = 80\sin{32\textdegree}

F_n = 42.39 \rm{N}

\cos{32\textdegree} = \dfrac{F_w}{80}

F_w = 80\cos{32\textdegree}

F_w = 67.84 \rm{N} math

The west-pulling dog would need to pull with 67.84 N and the north-pulling dog would need to pull with 42.39 N.

**Example 3**
Step 1, find the magnitude of vector //**A**// //**A**// = √ //Ax + Ay// //**A = √** 13²// + 17² //**A**// = √ //458// //**A** ~ 21.4// Step 2, find the direction

// Ɵ = tan^-1 ( 17 / 13 )// // Ɵ ~ tan^-1 ( 1.3077 )// // Ɵ ~ 52.6°// Vector //**A** has a magnitude of about 21.4 and a direction of 52.6°//

**Example 4**
Step 1, find the magnitude of vector //**A**//

//**A** = √ Ax + Ay// //**A** = √ 6² + 8²// //**A** = √ 100// //**A** = 10//

Step 2, find the direction

// Ɵ = tan^-1 (8 / 6) // // Ɵ = tan^-1 (1.33)// // Ɵ ~ 53.1°// Vector //**A**// has a magnitude of 10 at 53.1°

**Example 5**
Step 1, find the x components

//Ax = 5 cos(97°) ~ -0.609// //Bx = 8 cos(30) ~ 6.928// //Rx ~ -0.609 + 6.928 = 6.319//

Step 2, find the y components

//Ay = 5 sin(97°) ~ 4.963// //By = 8 sin(30°) ~ 4// //Ry ~ 4.963 + 4 = 8.963//

Step 3, find the magnitude of the resultant

//**R** = √ Rx² + Ry²// //**R** = √ 6.319² + 8.963²// //**R** ~ 10.967//

Step 4, find the direction

// Ɵ = tan^-1 (Ry / Rx) // // Ɵ // = tan^-1 ( 8.963 / 6.319) // Ɵ //= tan^-1 ( 1.418 ) // Ɵ ~ 54.816° // // The car had a magnitude of about 10.967 and direction //// 54.816°. //

**Example 6**
Step 1, find the x-axis components

//Ax = 9 cos(70°) ~ 3.078// //Bx = 6 cos(105°) ~ -1.553// //Cx = 4 cos(20°) ~ 3.758// //Rx ~ 3.078 + -1.553 + 3.758 = 5.283//

Step 2, find the y-axis components

//Ay = 9 sin(70°) ~ 8.457// //By = 6 sin(105°) ~ 5.795// //Cy = 4 sin(20°) ~ 1.368// //Ry~ 8.457 + 5.795 + 1.368 = 15.62//

Step 3, find the magnitude of vector **//R//** //**R** = √ Rx² + Ry²// //**R = √** 5.283² + 15.62²// //**R** ~ 16.489//

Step 4, find the direction

// Ɵ = tan^-1 (Ry / Rx) // // Ɵ = tan^-1 (15.62 / 5.283) // // Ɵ = tan^-1 ( 2.956 ) // // Ɵ ~ 71.31° // To finish your structure you will need a pipe cleaner that is 16.489cm long at an angle of 71.31°