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Chapter 12 – Vectors and the Geometry of Space12.2 – Vectors

12.2 – Vectors

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Definition - VectorThe term vector is used to

indicate a quantity that has both magnitude (length) and direction.

We denote a vector by a boldface letter (v) or by putting an arrow above the letter ( ).v

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NotationVectors always start from an

initial point called the tail and continue to the terminal point called the tip. We indicate this by writing v = . Vectors u and v are equal in direction and magnitude so u=v.

AB**************

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Definition - Zero VectorThe zero vector, denoted by 0,

has length zero. It is the ONLY vector with no specific direction.

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Definition – Vector Addition If u and v are vectors positioned so the

initial point of v is at the terminal point of u, then the sum u+v is the vector from the initial point of u to the terminal point of v.

Triangle Law Parallelogram Law

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VisualThis visual shows how the Triangl

e and Parallelogram Laws work for various vectors a and b.

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Try This On Your OwnHead to toe vector addition.

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Definition – Scalar Multiplication

If c is a scalar and v is a vector, then the scalar multiple cv is the vector whose length is |c| times the length of v and whose direction is the same as v if c>0 and opposite to v if c<0. If c=0 or v=0, then cv=0.

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More on ScalarsReal numbers work

like scaling factors.Two non-zero vectors

are parallel if they are scalar multiples of each other.

By the difference u – v of two vectors we mean

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Constructing u-vMethod 1 – Parallelogram Law

Draw v and –v and then add it to u.

Method 2 – Triangle Law

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ComponentsWe can place the initial point of a vector a at

the origin of a rectangular coordinate system. The terminal point of a has the coordinates of the form (a1,a2) or (a1,a2,a3) depending on if our coordinate system is a 2D or 3D one. The components are diagramed and written as follows:

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Definition – Position VectorThe position vector, , is the

representation of the vector from the origin to the point P.

OP**************

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Representing other vectorsGiven the points A(x1,y1,z1) and

B(x2,y2,z2), the vector a with representation isAB

**************

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MagnitudeThe magnitude or length is

denoted by |v| or ||v|| and obtained by the formulas:

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Combining VectorsAdding & Subtracting vectors and

multiplying a vector by a scalar.

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Properties of vectors

If a, b, and c are vectors in Vn and c and d are scalars, then

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Standard Base VectorsVectors i, j, and k are called the

standard base vectors. They have length 1 and point in the direction of the positive axis.

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Example 1 – pg. 777 #20

Find a+b, 2a+3b, |a|, |a-b| if

a = 2i – 4j + 4k

b = 2j - k

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Definition – Unit VectorA unit vector, u, is a vector

whose length is 1. For example, i, j, and k are all unit vectors.

a

ua

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Example 2 – pg. 799 # 23

Find a unit vector that has the same direction as the given vector.

-3i + 7j

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Example 3 – pg. 799 #30

If a child pulls a sled through the snow on a level path with a force of 50N exerted at an angle of 38o above the horizontal, find the horizontal and vertical components of the force.

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Example 4 – pg. 799 #34The magnitude of a velocity vector is called speed. Suppose the wind is blowing from the direction N45oW at a speed of 50 km/h. A pilot is steering a plane in the direction N60oE at an airspeed (speed in still air) of 250 km/h.

The true course, or track, of the plane is the direction of the resultant of the velocity vectors of the plane and wind. The ground speed of the plane is the magnitude of the resultant.

Find the true course and the ground speed of the plane.

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Example 5 – pg. 799 #38

The tension T at each end of the chain has magnitude 25 N. What is the weight of the chain?

12.1 – Three Dimensional Coordinate Systems

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More Examples

The video examples below are from section 12.2 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦Example 1◦Example 3◦Example 4

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Demonstrations

Feel free to explore these demonstrations below.

Head-to-Toe Vector Addition Vectors in 3D