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Vector Operations Chapter 3 section 2 A + B = ? B A

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Vector Dimensions When diagramming the motion of an object, with vectors, the direction and magnitude is described in x- and y- coordinates simultaneously. This allows vectors to be used for 1-d and 2-d motion.

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**How can I get to the red dot starting from the origin and can only travel in a straight line?**

x y

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There are 3 main different ways that I can travel to get from the origin to the red dot by only traveling in a straight lines. x y

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**Solving For The Resultant of 2 Perpendicular Vectors**

When two vectors are perpendicular to each other it forms a right triangle, when the resultant is formed. Right triangles have special properties that can be used to solve specific parts of the triangle. Such as the length of sides and angles.

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**(length of hypotenuse)²=(length of leg)²+(length of other leg)²**

Magnitude of a Vector To determine the magnitude of two vectors, the Pythagorean Theorem can be used As long as the vectors are perpendicular to each other. Pythagorean Theorem c²=a²+b² (length of hypotenuse)²=(length of leg)²+(length of other leg)²

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**Applied Pythagorean Theorem**

c2=a2+b R²=Δy²+Δx² (Mathematics) (Physics) Δx Δy R a b c

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Direction of a Vector To determine the direction of the vector, use the tangent function. Tangent Function Tanθ=opp/adj opp θ adj

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**Applied Tangent Function**

Δx Δy R a=opp b=adj c θ θ 𝑇𝑎𝑛 𝜃= Δ𝑦 Δ𝑥 (Physics) 𝑇𝑎𝑛 𝜃= 𝑜𝑝𝑝 𝑎𝑑𝑗 (Mathematics)

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**Recall Vector Properties**

Δx Δy R Δx Δy R = θ θ

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Example Problem A soldier travels due east for 350 meters then turns due north and travels for another 100 meters. What is the soldiers total displacement?

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Example Picture

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Example Work

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Example Answer R= °

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Vector Components Every vector can be broken down into its x and y components regardless of its magnitude or direction.

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**Vectors Pointing Along a Single Axis**

When a vector points along a single axis, the second component of motion is equal to zero.

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**Vectors That Are Not Vertical or Horizontal**

Ask yourself these questions. How much of the vector projects onto the x-axis? How much of the vector projects onto the y-axis?

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Components of a Vector x y A A x θ A x

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**Resolving Vectors into Components**

Components of a vector – The projection of a vector along the axis of a coordinate system. x-component is parallel to the x-axis y-component is parallel to the y-axis These components can either be positive or negative magnitudes. Any vector can be completely described by a set of perpendicular components.

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**Vector Component Equations**

Solving for the x-component of a vector. 𝐴𝑥=𝐴𝑐𝑜𝑠𝜃 (𝑥−𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑜𝑓 𝑣𝑒𝑐𝑡𝑜𝑟 𝐴 = 𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑣𝑒𝑐𝑡𝑜𝑟 𝐴 • cos(𝑎𝑛𝑔𝑙𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑣𝑒𝑐𝑡𝑜𝑟 𝐴) Solving for the y-component of a vector. 𝐴𝑦=𝐴𝑠𝑖𝑛𝜃

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Example Problem Break the following vector into its x- and y- components. A = °

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Example Problem Work A = °

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**Example Problem Answer**

Ax = 4.66 m/s Ay = 3.78 m/s

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Example Problem: A plane takes off from the ground at an angle of 15 degrees from the horizontal with a velocity of 150mi/hr. What is the horizontal and vertical velocity of the plane?

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Example Picture

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Example Work

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**Example Answer Horizontal velocity = 144.89 miles per hour**

Vx=144.89mi/hr Vertical velocity = miles per hour Vy=38.82mi/hr

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**Adding Non-Perpendicular Vectors**

When vectors are not perpendicular, the tangent function and Pythagorean Theorem can’t be used to find the resultant. Pythagorean Theorem and Tangent only work for two vectors that are at 90 degrees (right angles)

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**Non-Perpendicular Vectors**

To determine the magnitude and direction of the resultant of two or more non-perpendicular vectors: Break each of the vectors into it’s x- and y- components. It is best to setup a table to nicely organize your components for each vector.

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**Component Table x-component y-component Vector A - (A) Vector B - (B)**

Vector C - (C) Add more rows if needed Resultant - (R)

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**Non-Perpendicular Vectors**

Once each vector is broken into its x- and y- components : The components along each axis can be added together to find the resultant vector’s components. Rx = Ax + Bx + Cx + … Ry = Ay + By + Cy + … Only then can the Pythagorean Theorem and Tangent function can be used to find the Resultant’s magnitude and direction.

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Example Problem During a rodeo, a clown runs 8.0m north, turns 35 degrees east of north, and runs 3.5m. Then after waiting for the bull to come near, the clown turns due east and runs 5.0m to exit the arena. What is the clown’s total displacement?

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**Practice Problem Picture**

Step #1: Draw a picture of the problem

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Practice problem Work Step #2: Break each vector into its x- and y- components. x-component y-component Vector A - (A) Vector B - (B) Vector C - (C) Resultant - (R)

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**Step #3: Find the resultant’s components by adding the components along the x- and y-axis.**

x-component y-component Vector A - (A) Vector B - (B) Vector C - (C) Resultant - (R) +

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**Step #4: Find the magnitude of the vector by using the Pythagorean theorem. R2 = Δx2 + Δy2**

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**Step #5: Find the direction of the vector by using the tangent function. Tan θ = Δy/Δx**

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**Practice Problem Answer**

Step #5: Complete the final answer for the resultant with its magnitude and direction. Practice Problem Answer Resultant displacement = 57.21º

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