Scalar Quantities A scalar quantity (measurement) is one that can be described with a single number (including units) giving its size or magnitude Examples: Temperature, time, mass, distance. Can you think of some?

VectorsVectors A vector quantity is one that inherently requires magnitude and direction. Examples: Force, acceleration, displacement. Can you think of some?

Representing vectors When describing a vector it is useful to draw them. An arrow is used to create a graphical representation. The length of the arrow is proportional to the magnitude. The direction of the arrow corresponds to the direction of the vector.

Head to Tail Vector Addition If you have to add vectors you can’t just add the numbers to get the result. The reason is they are pointing in different directions 6 N 8 N 6 N + 8 N = 14 N

Measure angle Adding Vectors To add them we can simply move one so the head of one and tail of the other match up. This has to be done or our result will not be the right magnitude 6 N 8 N Copy angle Resultant Copy length

Analytical addition Although graphical methods work they require very precise drawings The analytical method uses trigonometry to add the vectors mathematically Lets review triangles

Meaning of sine, cosine & tangent The ratio of two sides of any right triangle with the same interior angles is always the same number independent of the size of the triangle, the triangle's orientation, or the units used to measure the sides of the triangles.right triangle There are three ways to form a ratio of the three sides of a triangle (six if you count the inverse ratios also). The symbols "sin", "cos", and "tan" are the tags/labels we use to identify which ratio is which.

ExampleExample Imagine a right triangle with the interior angles of 30, 60 & 90 Let’s look at the ratio of sides The sine would be ½ Click here for the unit circleClick here 1 2 √3√3 30

The Ratio The ratios are programmed into the calculator, and It’s really just using the unit circle

Co-linear and Perpendicular Co-linear vectors are on the same axis or number line Perpendicular vector simply make right angle with each other

Adding Perpendicular Vectors You can see by closing the tip of one and the tail of the other it makes a triangle For this we use the Pythagorean theorem The formula a 2 + b 2 = c 2 relates the three sides of a triangle

Vector components Sometimes vectors are not co-linear (on the same number line), or perpendicular For this we must find its “components” These are the x and y pieces that make up the right triangle using the vector as the hypotenuse

What?What? Look at the following vector. Imagine we shine a light from directly above it facing down. It will cast a shadow on the x axis equal to its length in the x direction That shadow is the x-component

What?What? Look at the same vector but now for the y-direction. Imagine we shine a light from the east side facing right. It will cast a shadow on the y axis equal to its length in the y direction That shadow is the y-component

Unit circle