Mayurbhanj School of Engineering,Baripada.� �VECTOR ALGEBRA�

PRESENTED BY-MR.ABHILASH MOHANTY

LECTURER IN MATHEMATICS

DEPARTMENT OF SCIENCE AND HUMANITIES

SCALAR

A SCALAR quantity

is any quantity in

physics that has

MAGNITUDE ONLY

Number value

with units

VECTOR

A VECTOR quantity

is any quantity in

physics that has

BOTH MAGNITUDE

and DIRECTION

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Vector quantities can be identified by bold type with an arrow above the symbol.

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V = 23 m/s NE

Vectors are represented by drawing arrows

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The length and direction of a vector should be drawn to a reasonable scale size and show its magnitude

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20 km

10 km

VECTOR APPLICATION

• ADDITION: When two (2) vectors point in the SAME direction, simply add them together.
• When vectors are added together they should be drawn head to tail to determine the resultant or sum vector.
• The resultant goes from tail of A to head of B.

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• Let’s Practice

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A man walks 46.5 m east, then another 20 m east.

Calculate his displacement relative to where he started.

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• VECTOR APPLICATION

SUBTRACTION: When two (2) vectors point in the OPPOSITE direction,

simply subtract them.

Let’s Practice some more….

A man walks 46.5 m east, then another 20 m west. Calculate his displacement relative to where he started.

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Graphical Method

Aligning vectors head to tail and then drawing the resultant from the tail

of the first to the

head of the last.

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Graphical Vector Addition A + B

Step 1 – Draw a start point

Step 2 – Decide on a scale

Step 3 – Draw Vector A to scale

Step 4 – Vector B’s tail begin at Vector A’s head. Draw Vector B to scale.

Step 5 – Draw a line connecting the initial start point to the head of B. This is the resultant.

NON CO-LINEAR VECTORS

When two (2) vectors are PERPENDICULAR to each other, you must use the PYTHAGOREAN THEOREM

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Let’s Practice

A man travels 120 km east then 160 km north. Calculate his resultant displacement.

• WHAT ABOUT DIRECTION?

In the example, DISPLACEMENT is asked for and since it is a VECTOR quantity,

we need to report its direction.

N

S

E

W

N of E

E of N

S of W

W of S

N of W

W of N

S of E

E of S

NOTE: When drawing a right triangle that conveys some type of motion, you MUST draw your components HEAD TO TOE.

N of E

• Directions

• There is a difference between Northwest and West of North

• NEED A VALUE – ANGLE!

Just putting N of E is not good enough (how far north of east ?).

We need to find a numeric value for the direction.

160 km, N

120 km, E

To find the value of the angle we use a Trig function called TANGENT.

So the COMPLETE final answer is : 200 km, 53.1 degrees North of East

• What are your missing components?

Suppose a person walked 65 m, 25 degrees East of North. What were his horizontal and vertical components?

65 m

25˚

H.C. = ?

V.C = ?

The goal: ALWAYS MAKE A RIGHT TRIANGLE!

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To solve for components, we often use the trig functions sine and cosine.

• Example

A bear, searching for food wanders 35 meters east then 20 meters north. Frustrated, he wanders another 12 meters west then 6 meters south. Calculate the bear's displacement.

35 m, E

20 m, N

12 m, W

6 m, S

-

=

23 m, E

-

=

14 m, N

23 m, E

14 m, N

The Final Answer: 26.93 m, 31.3 degrees NORTH of EAST

R

θ

• Example

A boat moves with a velocity of 15 m/s, N in a river which flows with a velocity of 8.0 m/s, west. Calculate the boat's resultant velocity with respect to due north.

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15 m/s, N

8.0 m/s, W

R_v

θ

The Final Answer : 17 m/s, @ 28.1 degrees West of North

• Example

A plane moves with a velocity of 63.5 m/s at 32 degrees South of East. Calculate the plane's horizontal and vertical velocity components.

63.5 m/s

32˚

H.C. =?

V.C. = ?

• Example

A storm system moves 5000 km due east, then shifts course at 40 degrees North of East for 1500 km. Calculate the storm's resultant displacement.

5000 km, E

40

1500 km

H.C.

V.C.

5000 km + 1149.1 km = 6149.1 km

6149.1 km

964.2 km

R

θ

The Final Answer: 6224.2 km @ 8.92 degrees, North of East