Republic Polytechnic

A114 Mathematics II

Worksheet 03: Points on the Same Plane

Discussion

1. Discuss the characteristics of a flat plane. How would a flat plane look like? Give some examples of a flat plane. Is define as a surface with no thickness.

2. What is a straight line? Give some examples of a straight line.
Y=mx+c, it’s a line where 2 points meet.

a) Given any two points in space, how would you find the vector from one point to another point?

b) What is the relationship of this vector with the straight line that passes through the two points?

3. What do you understand by the term “normal vector to a flat plane”? What is/are the angle/angle(s) between the normal vector and the flat plane?

Describe the relationship between any line on a plane with respect to the normal vector of the flat plane.

Exploring and Understanding 4.

Figure 1

Figure 1 above shows three points (i.e. A, B and C) lying on the same plane.

a) From the information given, find vectors , and . = (0 3 -5) = ( b) From what you have discussed in Q2 and Q3, what is the relationship between the normal vector to the flat plane and the three vectors , and ?

The normal vector is always perpendicular to the flat plane. c) What can you conclude about any vectors that lie on the plane with the normal vector to the plane? All perpendicular to the plane.

Dot Product 5. Similar to multiplication of two scalar quantities, we can also multiply two vectors. One of the ways to multiply two vectors is by taking the dot product of these two vectors as shown below:

If and , then a. *
Find the dot product between vectorand . -30 b. Given that vector e is zero vector, i.e., find the dot product of vector c and e, and vector d and e. What conclusion can you draw from here? All 0. The conclusion I can draw is that if any dot product of a zero vector will be zero

6. Figure 2 below shows two non-zero vectors a and b with angle between them.

a

b

Figure 2
It is given that the dot product of two vectorsis equal to where and are the magnitudes of a and b respectively.

If the two non-zero vectors are perpendicular to each other, what will be the value of?
Perpendicular to each other = 90 degrees.
Hence cos90.
Cos90 = 0
|a||b| x 0 = 0 = 0
(any dot product that is 0 is 90 degrees- perpendicular)

7. The normal vector n to the plane in the problem statement (also in Figure 1) is c. Find the dot product between vector and n. 0 d. Find the dot product between vector and n. 0 e. From your answer in part a) and part b), what can you conclude about the relationship between any vectors that lie on the plane and the normal vector to the plane? All the vectors that lie on the plane will have 90 degrees ,thus will have the dot product of 0. f. If you are given any point, how would you test whether the point lie on this plane? Use the dot product of the point, if it’s 0 (means it’s perpendicular to the plane), means its on the plane if its more or less means its not on the plane alr Vector Projection 8. A unit vector is defined as a vector of magnitude equals to 1. It can be found by dividing the vector by its own magnitude.
Find the unit vector of the normal vector to the plane in Figure 1.
Magnitude = 18.22
Unit vector = (0.768 0.549 0.329)
Magnitude of its unit vector is 0

9. Find vector using the coordinates of point A in Figure 1 and find the dot product between vector and the unit normal vector you have found in Q8. 5.812 (Note: O is the origin with coordinates (0,0,0))
From the definition of scalar product, we have

(is the unit vector of vector .) Using the definition above, interpret the geometric meaning of the dot product . (Hint: See Figure 3 bel

O

Figure 3

10. Find the dot product between vector (B is the point in Figure 3) and the unit normal vector of the plane. 5.811

Compare this value with the value you have obtained when you take the dot product of and the unit normal vector of the plane as in Q9.

Hence, make a guess of the value when you take the dot product of vector and the unit normal vector of the plane. Verify your guess by performing the calculation. 5.811

What conclusion can you draw from here? All dot product of the vector to the unit normal vector of the same plane is the same

Putting all together

11. Let us try to put together what you have learnt today, which your team will be sharing with the class during the presentation. You are to address the following in your sharing:

a. The steps taken to derive your answers b. The explanation to justify the decisions

Exploring Further – Equation of Plane

12. Based on your observation in Q4, now consider 2 points as shown in Figure 4 below, let point D represents any other points on the same flat plane as Point A, find an expression of vectorin terms of x, y and z. What is the relationship between vector and the normal vector to the flat plane?

Figure 4

13. Referring to Q7, what will be the value of dot product of vectorand the normal to the flat plane? Find the dot product between vector and the normal to the flat plane and use to it form an equation consisting of x, y and z.

What is the geometric interpretation of this equation?