PHYS 1501Q 1st Edition Lecture 4Outline of Last Lecture: Position, Velocity, and Acceleration ApplicationsI. Parachute Freefall Examplea. Find time of free fallb. Find velocity when chute opensc. Find total heightII. Relating Position to Velocity GraphsOutline of Current Lecture: Vector ManipulationsI. Vectorsa. Example Displacement VectorII. ScalarsIII. Vector Addition VisuallyIV. Vector Subtraction VisuallyV. Vector Multiplication Visually (Scalar)VI. Vector on a Coordinate SystemVII. Vector LengthVIII. Unit VectorsIX. Writing Vector in Terms of Unit ComponentsX. Vector Addition Using ComponentsXI. Example Vector Addition Calculation of CoordinatesThese notes represent a detailed interpretation of the professor’s lecture. GradeBuddy is best used as a supplement to your own notes, not as a substitute.XII. Vector Multiplication Overviewa. Scalarb. Dot Productc. Cross ProductCurrent Lecture: Vector ManipulationsI. Vectors- A vector is something that has a magnitude (length) and direction (where it points)- It can be represented visually as an arrow- Example Vector Quantities: Displacement, velocity, force- Vectors can be translated anywhere in the coordinate system (not rotated though)a. Example Displacement VectorObject moves from point A  BII. Scalars- A scalar is a single digit number not associated with a direction- Example Scalar Quantities: Mass, temperature, energyIII. Vector Addition Visually- Must have same units- Place tail of one vector to head of other vector. Draw a line connecting the disconnected head and tail. That line is the sum of the 2 vectors.o Commutative PropertyA + B = B + Ao Associative Property(A + B) + C = A + (B + C)IV. Vector Subtraction Visually- Must have same units- Must reverse direction of vector being subtracted. You can do this by flipping it 180 degrees or by multiplying it *(-1)- It’s the same as adding A + (-B)- A vector minus itself is zero B + (-B) = 0V. Vector Multiplication Visually (Scalar)- Multiplying a vector by a scalar changes the magnitude (length)- When multiplying you just add the vector to itself- Doesn’t change direction VI. Vectors on a Coordinate SystemA = magnitude (length)Θ = angle with respect to x-axisAx = AcosΘ (x-component)Ay = AsinΘ (y-component)VII. Vector Length- Length aka magnitude is never negative- Components Ax and Ay can be positive(+), negative(-), or 0A = √( Ax2+ A y2)VIII. Unit Vectors- i, j, and k are unit vectors (they are usually written with a triangle “hat” on, see picture)- Their magnitude is always 1- i, j, and k are NOT the same as x, y, and zo X Y Z are DIRECTIONS and I J K ARE UNITS along the corresponding axisThis is a 3D coordinate systemIf we used just 2D, it would be an X and Yaxis with i and j unit vectors respectivelyIX. Writing Vector in Terms of Unit ComponentsX. Vector Addition Using Components- Add x components to corresponding x components- Add y components to corresponding y components- Add z components to corresponding z componentsXI. Example Vector Addition Calculation of CoordinatesA = 1.0Θ = 30°Ax = 1.0cos(30°)  0.87Ay = 1.0sin(30°)  0.5B = 2.0Θ = 60°Bx = 2.0cos(60°) 1.0By = 2.0sin(60°) 1.73o Add together:(Ax + Bx) = Cx (Ay + By) = Cy(0.87+1.0) = 1.87 (0.5+1.73) = 2.23o Length of C vector:C = √(Cx2+C y2)A = √(1.872+2.232) = 2.91o Find angle of C vectorDo NOT add angles together you must useΘ = tan−1(CyCx)Θ = tan−1(2.231.87)Θ = 50°XII. Vector Multiplication Overviewa. Scalar- Multiply each element in vector by scalar- Multiplying by a scalar other than 1, will change the length of the vector- Multiplying by a negative number will make it go the opposite directionb. Dot Product- Two vectors multiplied together- Ex) used for workc. Cross Product- Two vectors multiplied together- Ex) Used for torque**Only scalar multiplication is required right now, more on other types of multiplication will be discussed in future