When you look at a plain sheet of paper, you would only take notice of its 2 dimensions, i.e. looking only at the length and breadth possibly because it is so flat. However, what happens when a box is now placed in front of you? Your vision seems to have upgraded to 3 dimensions because you are not just considering the length and breadth but the height or perhaps the thickness of the box. This article will explore 3-dimensional vectors.
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Jetzt kostenlos anmeldenWhen you look at a plain sheet of paper, you would only take notice of its 2 dimensions, i.e. looking only at the length and breadth possibly because it is so flat. However, what happens when a box is now placed in front of you? Your vision seems to have upgraded to 3 dimensions because you are not just considering the length and breadth but the height or perhaps the thickness of the box. This article will explore 3-dimensional vectors.
3-dimensional or 3D vectors are vectors that are represented on a three-dimensional plane or space to have three coordinates such as the x, y and z.
If we imagine a 3D plane with axis i, j and k, (which represents the x, y, and z-axis respectively) we can write a 3D vector as the sum of its i, j and k component.
Imagine a vector which travels from the origin (0,0,0) and goes to the coordinates (3,2,5). We could write that vector as
For this vector, the i component would be 3, the j component would be 2 and the k component would be 5.
The three-dimensional vector has three coordinates which are represented in the x, y and z-axis. Recall that in a two-dimensional plane, you have coordinates only on the x and y-axis. Thus, in a 2D vector coordinates are given in the form (x, y). However, the coordinates of 3D vectors are given in the form (x, y, z)
Begin by drawing a set of axis. Firstly, draw the vertical z-axis. Perpendicular to that, draw a y-axis. In between the z and y-axis, draw the x-axis. Note that all 3 axes are perpendicular to each other.
3-Dimensional axis (math.brown.edu)
After that, place a scale on each axis and mark the point where the head of the vector arrives. Then draw an arrow between the origin and the head of the vector. Finally, mark the coordinates of the head of the arrow.
Vector can also be written in matrix form. In this form, we can write the vector as three rows by one column matrix. The first row is the i component, the second row is the j component and the third row is the k component.
We do not write the x, y, and z terms in matrix form.
If we use the vector above as an example, we get:
We can combine two vectors to find the dot product of these vectors.
Suppose we have vector and vector , the dot product can be found by following the method below:
Step 1: Transpose vector , that is, convert it from a 3 rows by 1 column vector to a 1 row by 3 column vector.
For vector , vector
Step 2: Write the dot product of both vectors as the multiplication of both matrices.
Step 3: Perform the matrix multiplication:
Step 4: Simplify the matrix. You should end up with a 1-by-1 matrix.
Let vector , and vector . Find the dot product of vectors and .
Solution:
Writing both vectors in matrix form, we get:
and
Step 1:
Step 2:
Step 3:
Step 4:
Essentially, there are two main 3D equations. However, a third equation which is the angle between 3D vectors is derived from these two main equations. The two main equations are the dot product and the magnitude of a 3D vector equation.
For two certain 3D vectors A (x1, y1, z1) and B (x2, y2, z2) which are represented in the vector form
and
The dot product is
Find the product of Vector G and K located (-1, 2, 3) and (0, 5, 1) of a plane.
Solution:
By applying the dot product formula
Then,
The magnitude of a three-dimensional vector is derived using the extended Pythagoras theorem. Recall that the Pythagoras theorem is applied knowing the x and y-axis are perpendicular, note that the additional z-axis in 3D is perpendicular to both the x and y-axis. Hence, in order to calculate the magnitude of a certain 3D vector A (x1, y1, z1) which is represented in the vector form.
apply
Find the magnitude of vector C given by
Solution:
Since the magnitude of a vector is calculated as
Then the magnitude of vector C is
To find the angle between two corresponding 3D vectors, use the formula below:
Where is the angle between vectors a and b, is the dot product of vectors a and b, and where and are the magnitudes of vector a and vector b respectively.
Find the magnitude of the vector traveling from the origin to the coordinates (2,1,2).
Solution:
The vector can be written as
Using the equation above:
Therefore:
The magnitude of the vector is 3 units.
We can now combine all that we have learned to find the angle between two vectors!
Find the angle between vectors and vector .
Solution:
Writing the matrix form of these vectors:
and
Writing vector in transcript form:
Therefore:
The magnitude of vector is:
The magnitude of vector is:
Since:
Hence:
It is a line segment in three-dimensional space from Point A to Point B.
1. Write vector A in transpose form.
2. Write the dot product as a multiplication of the transposed matrix and the matrix for vector B.
3. Perform the vector multiplication.
4. Simplify.
Pythagoras' theorem.
Begin by drawing a set of axis. Firstly, draw the vertical z-axis. Perpendicular to that, draw a y-axis. In between the z and y axis, draw the x-axis. Note that all 3 axes are perpendicular to each other.
After that, place a scale on each axis and mark the point where the head of the vector arrives. Then draw an arrow between the origin and the head of the vector. Finally, mark the coordinates of the head of the arrow.
Generally, slope is only applicable to 2D vectors and as such the slope of vectors in 3D is undefined.
Begin by drawing a set of axis. Firstly, draw the vertical z-axis. Perpendicular to that, draw a y-axis. In between the z and y axis, draw the x-axis. Note that all 3 axes are perpendicular to each other.
After that, place a scale on each axis and mark the point where the head of the vector arrives. Then draw an arrow between the origin and the head of the vector. Finally, mark the coordinates of the head of the arrow.
3 dimensional vector problems are questions about 3D vectors which may require you to determine the magnitude of 3D vectors, angle between two 3D vectors or anything related to vectors in 3D.
What are 3D vectors?
3D vectors are vectors which are represented on a three dimensional plane or space to have three coordinates such as the x, y and z.
What are the coordinates of a 3D vector?
(x, y, z)
How do you plot a 3D vector?
Begin by drawing a set of axis. Firstly, draw the vertical z-axis. Perpendicular to that, draw a y-axis. In between the z and y axis, draw the x-axis. Note that all 3 axes are perpendicular to each other.
After that, place a scale on each axis and mark the point where the head of the vector arrives. Then draw an arrow between the origin and the head of the vector. Finally, mark the coordinates of the head of the arrow.
Can vectors be written in matrix form?
YES
P(1,2,-4) and Q(0,1,0) what is the dot product of P and Q.
2
Q(0,1,0), what is the magnitude of Q.
1
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