# 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 $\vec{A}$ and $\vec{B}$, draw the vector $\vec{A}$, then starting at the tip of $\vec{A}$ draw $\vec{B}$. The tail of $\vec{B}$ connects to the tip of $\vec{A}$.
• 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 the following vector additions graphically and numerically:

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

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

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

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

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

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

• 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 $\Delta \vec{d}_{block}$ that describes walking 1 block east.

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

iii. Find Tia's overall displacement.