Geometric Vectors A geometric vector is a representation of a vector - - PowerPoint PPT Presentation

42

5

1

0011 0010 1010 1101 0001 0100 1011

Culminating Review for Vectors

An Introduction to Vectors Applications of Vectors Equations of Lines and Planes Relationships between Points, Lines and Planes

An Introduction to Vectors Geometric Vectors

A geometric vector is a representation of a vector using an arrow diagram, or directed line segment, that shows both magnitude and direction.

Referred to as or Magnitude is or

30o A B 5 km

Directions may be represented as a bearing; which is a compass measurement where the angle is measured from the north in a clockwise direction.

30o N 50o N

quadrant bearing; which is a measurement between 0o and 90o east or west from the north-south line.

N60oE or 060o S40oW or 220o

Cartesian (Algebraic) Vectors We identify as a Cartesian vector because its endpoints can be defined using Cartesian coordinates.

position vector

In 3-space

Operations with Vectors

Scalar Multiplication Geometric Vectors Algebraic Vectors If Then If Then

Vector Addition Geometric Vectors Algebraic Vectors If and If and

Vector Subtraction Geometric Vectors Algebraic Vectors If and If and

Applications of Vectors

The Dot Product Geometric Vectors Algebraic Vectors If and We can use the dot product to solve for the angle between any two vectors:

The Cross Product Geometric Vectors Algebraic Vectors

Note: Direction is determined using the right-hand rule

If and OR 𝑏2 𝑏3 𝑏1 𝑏2 𝑐2 𝑐1 𝑐3 𝑐2 𝑦 𝑧 𝑨

down product minus the up product

The magnitude of the cross product is the area of a parallelogram defined by and

Scalar and Vector Projections The scalar projection of onto is The vector projection of onto is Recall:

Equations of Lines and Planes

Equations of Lines

Vector Equation:

𝑦 = 𝑦𝑝 + 𝑢𝑏

If you equate the respective x, y and z components we

• btain the parametric equations:

𝑧 = 𝑧𝑝 + 𝑢𝑐 𝑨 = 𝑨𝑝 + 𝑢𝑑

𝑦 = 𝑦𝑝 + 𝑢𝑏

If you isolate each parametric equation for the parameter, 𝑢, we have

𝑧 = 𝑧𝑝 + 𝑢𝑐 𝑨 = 𝑨𝑝 + 𝑢𝑑

Combining these statements gives the symmetric equation of a line:

Equations of Planes

Vector Equation:

x = xo + sa1 + tb1

If you equate the respective 𝑦, 𝑧 and 𝑨 components we

• btain the parametric equations:

y = yo + sa2 + tb2 z = zo + sa3 + tb3

[x, y, z] = [xo, yo, zo] + s[a1, a2, a3] + t[b1, b2, b3]

Cartesian Equation:

The Cartesian (or scalar) equation of a plane is of the form 𝐵𝑦 + 𝐶𝑧 + 𝐷𝑨 + 𝐸 = 0 with normal 𝑜 = 𝐵, 𝐶, 𝐷 . The normal 𝑜 is a vector perpendicular to all vectors in the plane. Given two directions vectors the normal to a plane is determined by calculating the cross product of the direction vectors.

Relationships between Points, Lines and Planes

The Intersection of Two Lines l1 : [x, y, z] = [–3, 1, 4] + s[–1, 1, 4] l2 : [x, y, z] = [1, 4, 6] + t[–6, –1, 6] x = –3 – s y = 1 + s z = 4 + 4s x = 1 – 6t y = 4 – t z = 6 + 6t Convert to parametric form Select any two of the three equations and equate them l1 l2 –3 – s = 1 – 6t 1 + s = 4 – t Solve for s and t and sub these results into both lines to check your solution

The Intersection of a Line and a Plane l : [x, y, z] = [3, 1, 2] + s[1, –4, –8]  : 4x + 2y – z – 8 = 0 x = 3 + s y = 1 – 4s z = 2 – 8s Convert to line to parametric form Equate 4(3 + s) + 2(1 – 4s) – (2 – 8s) – 8 = 0 Solve for s sub this result into the line to determine the point of intersection

The Intersection of Two Planes 1 : x – y + z = 3 2 : 2x + y – 2z = 3 Analyze the normals; what do they tell you? Eliminate one of the variables, for example, z, 21 + 2 gives 4𝑦 − 𝑧 = 9 Isolate for a variable 𝑧 = 4𝑦 − 9 Introduce a parameter Let 𝑦 = 𝑢 𝑧 = 4𝑢 − 9 Substitute x = t and y = 4t – 9 into one of the planes t – (4t – 9) + z = 3 z = 3t – 6 The line of intersection is x = t y = 4t – 9 z = 3t – 6

The Intersection of Three Planes 1: 2𝑦 + 𝑧 − 𝑨 = −3 2 : x – y + 2z = 0 3 : 3x + 2y – z = –5 Create two new equations 21 + 3 : 5x + y = –6 1 : 2x + y – z = –3 3 : 3x + 2y – z = –5 21 : 4x + 2y – 2z = –6 2 : x – y + 2z = 0 1 – 3 : –x – y = 2 Solve for x and y, and use these values and one of the planes to determine the value of z. Verify the results using the other planes.