Vectors and Vector Addition

Vectors vs. Scalars Scalars are physical quantities having only magnitude– that is, a numerical value & units. Ex: a speed of 4 m/s Distance and speed are scalars. Vectors are quantities having magnitude and direction. Ex: a velocity of − 4 m/s or 4 m/s, up or 4 m/s, right or 4 m/s, west. Displacement, velocity, acceleration, and force are vectors.

Scalar Operations Scalars can be added or subtracted as long as they have the same units. You cannot add a speed and a distance! Scalars can multiplied and divided. The resulting magnitude will have different units from the original scalars. For example, dividing distance(m) by time(s) gives an answer in units of m/s.

Vector Operations Vectors can be added and subtracted as long as they have the same units. You will learn how to do this. Vectors can be multiplied, however, that process is beyond the scope of this regular physics course since it involves dot and cross products.

Vector Addition You will learn to add vectors graphically. That means you will represent each vector by an arrow whose scaled length represents the magnitude of the vector quantity and whose pointed end (the head) indicates the direction. Vectors have both a head and a tail. HeadTail

Vector Addition: How? In most cases, vectors do not add like ordinary numbers! For the purpose of adding vectors, you may move the arrows around as long as you do not change their scaled length or direction. Vectors will be added graphically by placing the tail of one vector on the head of the other vector. TAIL to HEAD.

Vector Addition: Example 1 Add these two velocity vectors: v 1 = 5 m/s, west and v 2 = 12 /ms, south Pick a scale to draw them, such as 2 m/s = 1 cm (Scale) v 1 = (2.5 cm long) v 2 = (6.0 cm long) v1v1 v2v2

Example 1 (cont’d)—graphical method Redraw them tail to head. The order makes no difference! Draw in the resultant vector v R (representing the vector sum) by completing the triangle. The resultant starts from the tail of the first vector and ends at the head of the last vector: START to END. vRvR v1v1 v2v2

Example 1 (cont’d)—graphical method After drawing the resultant, measure it with a ruler. Reapply the scale to work backwards from cm on your paper to actual m/s. In this example, the resultant arrow indicates a resultant velocity of 13 m/s in the direction of southwest (SW).

Solving Example 1—algebraic method Since the vectors v 1, v 2, and v R form a right triangle, you can also solve this problem by using the Pythagorean Theorem: a 2 + b 2 = c 2 Here, v v 2 2 = v R = = 169 v R = square root of 169 = 13 m/s

Example 2 : Adding 3 vectors Find the resultant d R by adding these displacement vectors: d 1 = 2 m, east ; d 2 = 3m, north; and d 3 = 4 m, west. A possible scale to use to draw these is: 1m = 1 cm

Example 2 (cont’d) Draw them tail to head and again tail to head. Then draw in the resultant displacement. Measure the resultant and apply the scale. It measures 3.4 cm. That represents d R = 3.4 m, NW dRdR

Example 3: collinear vectors Collinear vectors are vectors that lie along the same line. Only collinear vectors add or subtract like ordinary numbers or scalars. Add two force vectors: F 1 = +25 N and F 2 =+15 N F 1 + F 2 = +40 N = F R F1F1 F2F2 FRFR

Example 4: Adding opposite vectors Vectors are also collinear if they lie along a straight line in opposite directions. In this case, adding two vectors effective- ly means they subtract. The resultant will be in the direction of the longer one. Add F 1 = 25 N, right to F 2 = 15 N, left. 25 N – 15 N = +10 N F1F1 F2F2 F R = 10 N, right(+) +–