Vector Defined

• Scalar - a quantity that just has a number and unit

• Vector - a quantity that represents a number, units, and direction

  • number and unit is referred to as magnitude

  • Examples

    • Displacement - 3.0 ft left

    • Velocity - 50 mph east

    • Acceleration - 322 ft/s² down

    • Force - +200 lbs.

Ways Vectors are represented

• A vector sign above the symbol

  • Force - F^→

  • Acceleration - A^→

• A bold print symbol

  • Velocity - V

  • Displacement - d

• An arrow tipped line segment

  • The length of the line, drawn to scale, indicates the magnitude of the vector

  • The direction of the arrow tip is the direction of the vector quantity

The Resultant

• A^→ + B^→ = R^→

• The result is a vector called the resultant vector

  • Draw the resultant from the tail of the first vector to the head of the last vector

  • The resultant is a vector; it has magnitude (#, Unit) and direction

Adding Vectors Graphically

• A^→ + B^→ =

• Add the vectors by placing the tail of the second vector to the head of the first vector

• If more than two vectors, place the tail of the next vector to the head of the last vector drawn

Graphical Method Steps

• Indicate directions on paper for every problem (compass)

• Change vector to a scale unit

  • 5 lbs. = 5 cm on paper

• Draw original axis (solid lines)

• Draw 1st vector to scale, by placing the tail at the origin

• Draw 2nd axis at the head of 1st vector (dotted axis)

  • Parallel to original axis

• Draw 2nd vector to scale, from dotted axis

• If more than 2 vectors, repeat the steps

• Draw resultant vector from origin, tail of 1st vector to the head of last vector

• Measure length of the resultant with ruler

  • Length and Magnitude

• Measure angle with a protractor for the direction of the resultant

  • Direction

  • Measured from origin

Vector Resolution

• Breaking a vector into an x, y, and z component

  • For AP Physics, z will always be 0

• All vectors have x, y, and z components

  • Sometimes they are 0

Graphical Vector Resolution

• Write down given vector

• Indicate directions

• Scale vector

• Draw original axis

• Draw vector to scale

• Extend x axis and draw perpendicular from x axis to tip of vector

  • R_y

• Draw from origin to perpendicular

  • R_x

• Measure R_x and R_y

• Convert back to original units

• Indicate directions

Vector Resolution Mathematical Method

• Write down the given vector

• Indicate the directions

• Draw original axis

• Sketch vector

• Extend x axis and draw arrow perpendicular from x axis to tip of vector (R_y)

• Draw from origin out to perpendicular (R)

• Use sin(x) and solve for opposite side (A_y)

• Use cos(x) and solve for adjacent side (A_x)

Addition

• Write down the given vector

• Indicate the directions

• Sketch each vector on separate axis

• Resolve each vector using trig vector resolution

• A_x + B_x = R_x || A_y + B_y = R_y

• Use Pythagorean theorem to find magnitude of R

• Use tangent to find angle and direction