Vectors are closely related to Mathematics and we also use them a lot to make our daily lives easier. There are also several types of vectors, starting from 2-dimensional vectors and there are also 3-dimensional vectors.
What is a 3D Vector
In this 3D vector material, you need to know first about its meaning. Vectors in 3D or spatial structures are vectors that have 3 axes, namely X, Y and Z. These axes are perpendicular to each other and have interesting intersections of the three axes used as their bases.
It is important to understand before knowing how to draw 3D vectors from 3D vector images, the vectors in space are vectors that are located in 3D space. This space can later intersect perpendicularly. Then the result of the intersection is O and the point O is called the central axis.
As for the explanation regarding the rules related to three-dimensional vectors, you can check them all directly below!
- The middle finger shows the X axis, numbers whose direction is parallel to the middle finger and which are located after O fall into the positive number category. Then the opposite direction and location are negative numbers.
- The thumb indicates the X axis, this number will later be in the direction of the thumb and is located after O to be a positive number. Then for the opposite position and direction, it becomes a negative number.
- The index finger indicates the presence of the Y axis. The numbers that are located after O and the direction is according to the index finger are numbers with positive values. Furthermore, for the direction and location, it can be called a negative number.
Finding the Length of a 3-Dimensional Vector
What is meant by vector length? Vector length is the distance from the starting point to the end point of the vector. This is in accordance with the explanation from the algebraic side which explains that the starting point of the vector and the end point are indeed in the form of coordinates in terms of both two and three dimensions. So the formula for determining the length of the vector can be seen from the distance of two points.
Do you know that the presentation of a vector can be given in the form of a single-column matrix, a pair of ordered numbers or a combination of its unit vectors. For a 3-dimensional vector, the starting point is O(0, 0, 0) then for the end point is (p, q, r). While for writing it as a vector v = (p, q, r).
The length of the vector itself is symbolized by lowercase letters flanked by straight signs like absolute values. Then the magnitude of the vector length value is the distance from the end of the vector to the base of the vector. Later, for the length of the vector whose position u and v can be obtained through the calculation of the root of the numbers that make up the position of the vector.
3D Vector Example Questions
You need to know before understanding the addition of 3-dimensional vectors and how to calculate 3-dimensional vectors, it is important to know the following equations first. If the location of the coordinates of the end of the vector and the base of the vector is known, then its length can be calculated using the Pythagorean formula. For example, if the vector AB is located at A(x1, y1).
While the other condition is that the end of the vector is in the B(x2, y2) section. This causes the intersection of the projection lines of the two vector points AB to form a right triangle which provides an equation for calculating the length of the vector AB, namely |AB|2 = (x2 – x1)2 + (y2 – y1)2. Then what about the examples of questions that are often encountered?
As for some questions that you can study directly, just check them all out below!
Example Question 1
Three vectors A = (2, 5), B = (-3, 7), and C = (1, -2) form a triangle. Calculate the length of the sides of the triangle and the size of the angle formed at point A.
Answer:
Length of side AB:√[(−3−2)2+(7−5)2]=√(25+4)=√29
Length of side AC:√[(1−2)2+(−2−5)2]=√(1+49)=√50
Example Question 2
Given vector A = (4, -2) and vector B = (1, 3). Calculate the scalar product of the two vectors.
Answer:
Scalar product of A and B:A•B=(4×1)+(−2×3)=4−6=−2
Example Question 3
There is a vector X that has a magnitude of 10 units. Based on the data, what should the vector -X be?
A. Vector -X must have a magnitude of -10 units and the same direction as vector X.
B. Vector -X must have a magnitude of 10 units and the same direction as vector X.
C. Vector -X must have a magnitude of 10 units and a direction opposite to vector X.
D. Vector -X must have a magnitude of 10 units and a direction perpendicular to vector X.
E. Vector -X must have a magnitude of -10 units and a direction perpendicular to vector X.
Answer: C. Vector -X must have a magnitude of 10 units and a direction opposite to vector X.
Discussion: If a vector quantity is written as -X, it means that its direction is opposite to vector X. However, its magnitude is the same, meaning it does not change, namely the same as vector X.
Example Question 4
An airplane experiences a change in position from point P(4, 2) to point Q(8, 10). Determine the displacement vector of the airplane.
Answer:
Displacement vector from P to Q: PQ = (8 – 4, 10 – 2) = (4, 8).
Example Question 5
Suppose 𝐴 = (1,2) and 𝐵 = (3,4), then 𝑢⃗ is the position vector of 𝑂𝐴 and 𝑣
is the position vector (𝑂𝐵)
Note that
𝑢⃗ + 𝑣 = 𝐴 + 𝐵
= (1,2) + (3,4)
= (4,6)
On the other hand, since 𝑢⃗ and 𝑣 have the same initial, namely 𝑂, the parallelism rule applies.
parallelogram, as a result 𝑢⃗ + 𝑣 = 𝑣 + 𝑢⃗ is also the position vector 𝑂𝐷 where 𝐷 = (4,6).