Vectors What is a vector?

What is a vector? In science measurements of various quantities can be categorized into two main groups Scalar and Vector. Quantities that have only magnitude (size) are called Scalars. Quantities that have both magnitude AND direction are called Vectors. Magnitude – a number, quantity, how much, size

Scalar Vector Distance – how far Speed – how fast Displacement – how far with a direction Velocity – how fast with a direction Acceleration – speeding up/slowing down with a direction

Scalar Vector Mass – the amount of matter in an object Time – how long Volume – the amount of space an object takes up Work – Force acting through a displacement Weight – the force of gravity acting on an object Force – a push or pull Torque – force of rotation Magnetic Field Strength

How to represent a vector? All vectors in science are represented by Arrows The size represents the magnitude The arrow represents the direction of the vector Negatives mean opposite direction

Adding/Subtracting (adding a negative) When vectors are 0 and 180 degrees in difference You add vectors just like you would numbers, but you must take their direction into account Ex: 5 S + 10 S = 15 S

Drawing Method When visually representing vectors you must make sure you follow the “Tip to Tail” method. Draw your first vector then draw the next vector at the tip of last one. Then solve it mathematically. The answer is called the resultant vector (VR). tail tip Vector

Drawing Method Now you try! 3.0 West + 4.0 West = 4.0 North + 5.0 South = 6.0 North – 3.0 South = HW: Diagram Skills front and Vector Addition 1-6

Adding to vectors that are 90 degree difference Before we start this procedure we must first review our trigonometric functions. Remember SOH CAH TOA  θ =

Adding Vectors with 90 cont’d First DRAW your vector addition using the “Tip to Tail” method Then use the Pythagorean Theorem and the Inverse Tangent function to solve mathematically. Ex: Add 10.0 m/s E to 15.0 N Step 2 (Pythagorean) Step 3 (Tan-1) HW: Vector Addition Practice 1-10

Now let’s try and do the oppostie! If I give you a vector at an angle and ask for the x and y components of the vector, that means I’m giving you the resultant and asking for the vectors that make it up. Ex: What are the components of 25 m, 65° NE? HW: Vector Addition Practice 11-15

Adding With Multiple Vector Types There are five steps: Draw a picture Resolve any vectors at an angle into their x and y components Find your x sum and y sum by combining all the x’s and all the y’s Draw the new picture using your x sum and y sum Use the Pythagorean Theorem and the tangent function to solve

Let’s Try One! 16.0 m East + 20.0 m, 35.0° SE HW: Vector Addition Practice 16-20

Divide/Multiply a Vector by a Scalar The rule here is just divide/multiply the magnitude and keep the direction the same. Make sure you keep your sig. fig. rules Example: 10.0 m S / 4.0 = 2.5 m 3.0 m E x 4.0 m = 12 m E

Multiply a Vector by a Another Vector There are 2 types of vector multiplication. DOT PRODUCT In this calculation when you multiple the vectors you get a SCALAR for the answer.   CROSS PRODUCT In this calculation when you multiple the vectors you get a vector as the result. However, you have to use the right hand rule to find the resulting direction.