# Vectors PPT ### Magnitude

• The magnitude of a vector is represented by its length.

• You can multiply the magnitude of a vector by a scalar quantity to change its length.

A = 10m

0.5A =5m

2A =20m

### Direction

• All vectors must have a direction.

– North, south, east, west

– Or an angle

• Sometimes measured from the horizontal or vertical

• Sometimes measured from the positive x-axis and around to 360 o

230 o

30 o

40 o

• Must manipulate all vectors so they are put “tip to tail”

30 o

A = 10m

40 o

• Resultant points from start to finish.

### Sketch the vector sum of A+B

• Must manipulate all vectors so they are put “tip to tail”

• Resultant points from start to finish.

40 o

A = 10m

### Sketch the vector sum of A+B+C

• Must manipulate all vectors so they are put “tip to tail”

• Resultant points from start to finish.

30 o

A = 10m

40 o

### Sketch the resultant of A-B

• Flip the direction of the one being subtracted

A = 10m

40 o

### Components of Vectors

• Every vector will have a…

– Horizontal component that points directly left or right

– Vertical component that points directly up or down.

• The components should be drawn tip to tail and lead to the same point as the original vector.

• The angle goes by the start.

𝜽

Horizontal component

### Calculating Vector Components

• In this case…

– Horizontal component is the adjacent side which can be calculated using 𝐴 = 𝐴 cos 𝜃 = 25m cos40 o = 19m 𝑥

– Vertical component is the opposite side which can be calculated using 𝐴 𝑦

= 𝐴 𝑠𝑖𝑛 𝜃 = 25m sin 40 o = 16m

• REMEMBER: horizontal is not always cosine. It depends where the angle is located!

– If the angle is to the vertical, sine and cosine would flip.

𝜽 = 𝟒𝟎 𝒐

Horizontal component

Find the components of the following vectors

A = 10m

𝑨 𝒙

𝑨 𝒚

= +𝟏𝟎𝒎

= 𝟎 m

30 o

40 o

𝐴 𝑥

= 𝐴 cos 𝜃

= 15m cos40 o

= 11.5m

𝐴 𝑦

= 𝐴 𝑠𝑖𝑛 𝜃

= 15m sin 40 o

= 9.6m

𝐴 𝑥

= 𝐴 sin 𝜃

= 20m sin 30 o

= 10.0 m

𝐴 𝑦

= 𝐴 𝑐𝑜𝑠 𝜃

= 20m cos 30 o

= 17.3m

### Find the magnitude and direction of A+B

• Step 1: find the components of each vector

• Step 2: add all of the x-components together to find the resultant’s xcomponent

• Step 3: add all of the y-components together to find the resultant’s ycomponent

• Step 4: build your actual resultant out of its components you just fund.

• Step 5: use Pythagorean Theorem and SohCahToa to find magnitude and direction of resultant.

A = 10m

𝑨 𝒙

𝑨 𝒚

= +𝟏𝟎𝒎

= 𝟎 m 𝑎 2

12.5

2

+ 𝑏 2

+ 9.6

= 𝑐

2

2

= 𝑐 2 c= 15.8m

tan 𝜃 =

𝐴 𝑦

= 𝐴 𝑠𝑖𝑛 𝜃

= 15m sin 40 o

RESULTANT

X-component:

10m+11.5m = 21.5m

Y- component:

0m+9.6m = 9.6m

𝐴 𝑥

= 21.5𝑚

𝐴 𝑦

= 9.6m

40 o

𝐴 𝑥 tan 𝜃 =

= 𝐴 cos 𝜃

12.5

𝜽 = 𝟑𝟕. 𝟓 𝒐

= 15m cos40 o

= 11.5m