Simple vector arithmetic question

In summary, The conversation is discussing the relationship between three vectors, a, b, and c, where a and c have an angle of 45 degrees between them. The question is whether b, which is generated by subtracting a from c, will be orthogonal to a. After using MATLAB to sketch and generate random examples, it is determined that b is not orthogonal to a in every case. However, it is later realized that a mistake was made in the calculations. The final statement states that the three vectors will form a triangle with angles of 45 degrees and 135 degrees.
  • #1
The_Engineer
18
0
I have two vectors: a = <ax, ay, az> and c = <cx, cy, cz>

which have an angle of 45 degrees between them.


If I get another vector by b = c - a then shouldn't b be orthogonal to a? I'm assuming this since a + b = c
 
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  • #2
Did you try to sketch some vectors and to come up with a counter example?
 
  • #3
A.T. said:
Did you try to sketch some vectors and to come up with a counter example?

Yes I have been using MATLAB to sketch and generate random examples. Here is one...

a = <0.3814, 0.9023, 0.2010>
c = <0.3965, 0.7378, -0.5463>

The angle between these vectors is 45 degrees.

I want a vector b such that b is orthogonal to a AND 45 degrees from c. Graphically speaking, this means a + b = c

Solving for b,
b = c - a = <0.0151, -0.1645, -0.7473>.



But b dot a ≠ 0 therefore they aren't orthogonal. Why aren't a and b orthogonal?


This happens for every random sample I make, starting with 2 vectors that are 45 degrees apart.
 
  • #4
Nevermind I figured it out... I made a mistake by normalizing the resultant, which in turn messed up my final answer.
 
  • #5
The vectors you describe will form a triangle. With one angle at 45 deg. the other two angles could be anything that add up to 135 deg.
 

1. What is vector arithmetic?

Vector arithmetic is the mathematical operation of adding, subtracting, multiplying, and dividing vectors. Vectors are mathematical quantities that have both magnitude (size) and direction. In vector arithmetic, we can perform these operations on vectors to manipulate their magnitude and direction.

2. What are the basic operations in vector arithmetic?

The basic operations in vector arithmetic are addition, subtraction, scalar multiplication, and dot product. Addition and subtraction involve combining or separating vectors, while scalar multiplication involves multiplying a vector by a number. The dot product is a type of multiplication that results in a scalar (a single number) rather than a vector.

3. How do you add or subtract vectors?

To add or subtract vectors, we use the parallelogram rule. This involves placing the tail of one vector at the head of the other and drawing a new vector from the tail of the first vector to the head of the second vector. The resulting vector is the sum or difference of the original vectors, depending on whether we are adding or subtracting.

4. What is scalar multiplication in vector arithmetic?

Scalar multiplication is the process of multiplying a vector by a single number (scalar). This operation changes the magnitude of the vector but not its direction. If the scalar is positive, the resulting vector will have the same direction as the original vector. If the scalar is negative, the resulting vector will have the opposite direction.

5. What is the dot product in vector arithmetic?

The dot product, also known as the scalar product, is a type of multiplication between two vectors that results in a scalar (single number) rather than a vector. It is calculated by multiplying the corresponding components of the two vectors and then adding them together. The dot product is useful for finding the angle between two vectors and for determining if two vectors are perpendicular.

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