Vector operations in one dimension

Vector operations in one dimension

Lessons

In this lesson, we will learn:
  • How to add, subtract, multiply, and divide vectors
    • Graphically (with diagrams)
    • Numerically (with math)

Notes:

Just like scalars, vectors can be added, subtracted, multiplied, and divided:

  • When vectors are added, the vector diagram is drawn by tip-to-tail addition. To add A\vec{A} and B\vec{B}, draw the vector A\vec{A}, then starting at the tip of A\vec{A} draw B\vec{B}. The tail of B\vec{B} connects to the tip of A\vec{A}.
  • Taking the negative of a vector "flips" the vector to point in the opposite direction, while keeping the same magnitude.
  • Multiplying or dividing a vector by a positive scalar changes the magnitude of the vector, while keeping the same direction.
  • Introduction
    Introduction to vector operations:
    a)
    What is the difference between adding scalars and vectors?

    b)
    Solving vector problems graphically

    c)
    Solving vector problems numerically

    d)
    Sign convention for vector problems


  • 1.
    Solve vector addition problems
    Solve the following vector additions graphically and numerically:

    i. A+B=C\vec{A} + \vec{B} = \vec{C}, if A=10m\vec{A} = 10 m [E], B=7m\vec{B} = 7 m [E].

    ii. v1+v2=vres\vec{v}_{1} + \vec{v}_{2} = \vec{v}_{res} if v1=4.5m/s\vec{v}_{1} = 4.5 m/s [E], v2=13.2m/s\vec{v}_{2} = 13.2 m/s [W]


  • 2.
    Solve vector subtraction problems
    Δd1=1.2km\Delta \vec{d}_{1} = 1.2 km [N], Δd2=0.8km\Delta \vec{d}_{2} = 0.8 km [N]. Solve the equation Δd1Δd2=Δdres\Delta \vec{d}_{1} - \Delta \vec{d}_{2} = \Delta \vec{d}_{res} graphically and numerically.

  • 3.
    Solve vector multiplication and division problems
    D=12m/s \vec{D} = 12 m/s [W]. Solve the following graphically and numerically:

    i. 3D=E3\vec{D} = \vec{E}

    ii. -D2=F\frac{\vec{D}}{2} = \vec{F}


  • 4.
    Create Vector Equation and Diagram from Word Problem
    A city block is 50 m long. While doing errands, Tia walks 1 block east, then three fifths of a block west, then two blocks east.

    i. Write the displacement vector Δdblock\Delta \vec{d}_{block} that describes walking 1 block east.

    ii. Describe this situation with a vector equation and a vector diagram in terms of Δdblock\Delta \vec{d}_{block}

    iii. Find Tia's overall displacement.