Civil Engineering (CE) Exam > Civil Engineering (CE) Videos > Engineering Mechanics > Cross Product of Vectors
Cross Product of Vectors Video Lecture | Engineering Mechanics - Civil Engineering (CE)
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24 videos|59 docs|53 tests
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FAQs on Cross Product of Vectors Video Lecture - Engineering Mechanics - Civil Engineering (CE)
| 1. What is the definition of the cross product of vectors? |  |
The cross product of two vectors is a vector that is perpendicular to both of the original vectors and has a magnitude equal to the product of the magnitudes of the original vectors multiplied by the sine of the angle between them. It is denoted by the symbol "×" or "⨯".
| 2. How is the cross product of vectors calculated? | |
To calculate the cross product of two vectors A and B, you can use the determinant method. Arrange the components of the vectors in a 3x3 matrix, where the first row consists of the unit vectors i, j, and k, and the second and third rows contain the components of vectors A and B, respectively. Then, calculate the determinant of this matrix. The resulting vector will be the cross product of A and B.
| 3. What is the geometric interpretation of the cross product? | |
The cross product of two vectors has a geometric interpretation as the area of the parallelogram formed by the two vectors. The direction of the cross product vector is perpendicular to the plane containing the original vectors, and its magnitude represents the area of that parallelogram.
| 4. What are some applications of the cross product of vectors? | |
The cross product has various applications in physics and engineering. Some examples include calculating torque, determining the direction of magnetic fields, analyzing fluid dynamics, solving problems related to angular momentum, and understanding the motion of objects in three-dimensional space.
| 5. Are there any special properties of the cross product? | |
Yes, the cross product of two vectors has several special properties. These include the distributive property, the fact that the cross product of two parallel vectors is zero, and the right-hand rule, which determines the direction of the resulting vector. Additionally, the magnitude of the cross product is equal to the product of the magnitudes of the original vectors multiplied by the sine of the angle between them.
Text Transcript from Video
Suppose we have two arrows.
Let us define the cross product of these two arrows to be a third arrow.
The arrow representing the cross product
is always exactly 90 degrees to the two original arrows.
The length of the arrow representing the cross product
is always exactly equal to the area of the parallelogram
that is formed by the original two arrows.
The cross product of two arrows plays a critical role
in many areas of science and engineering.
Another type of calculation that plays a critical role
is what we refer to as the dot product of two arrows.
The first arrow defines a line.
This line and the second arrow form a triangle.
Consider the length of this side of the triangle.
Now, consider this length multiplied by the length of the first arrow.
The result of this multiplication is what we refer to as the dot product.
If the first arrow and the red arrow are pointed in opposite directions,
then the dot product is negative.
Whereas the cross product of two arrows is another arrow,
the dot product of two arrows is just a number.
Whenever we take the dot product of two arrows, we will
always get the same answer regardless of
which of the two arrows we refer to as the first arrow,
and which of the two arrows we refer to as the second arrow.
This is not the case with cross product.
If we reverse the order of the two original arrows,
then the arrow for the cross product will end up pointing in the opposite direction.
There is a rule for using your right hand to remember
the direction in which the arrow representing the cross product is pointing.
If you point your right hand in the direction of the first arrow,
and then curl your fingers in the direction of the second arrow,
your thumb will end up pointing in the direction of the cross product.
Much more information is available in the other videos on this channel,
and please subscribe for notifications when new videos are ready.
Let us define the cross product of these two arrows to be a third arrow.
The arrow representing the cross product
is always exactly 90 degrees to the two original arrows.
The length of the arrow representing the cross product
is always exactly equal to the area of the parallelogram
that is formed by the original two arrows.
The cross product of two arrows plays a critical role
in many areas of science and engineering.
Another type of calculation that plays a critical role
is what we refer to as the dot product of two arrows.
The first arrow defines a line.
This line and the second arrow form a triangle.
Consider the length of this side of the triangle.
Now, consider this length multiplied by the length of the first arrow.
The result of this multiplication is what we refer to as the dot product.
If the first arrow and the red arrow are pointed in opposite directions,
then the dot product is negative.
Whereas the cross product of two arrows is another arrow,
the dot product of two arrows is just a number.
Whenever we take the dot product of two arrows, we will
always get the same answer regardless of
which of the two arrows we refer to as the first arrow,
and which of the two arrows we refer to as the second arrow.
This is not the case with cross product.
If we reverse the order of the two original arrows,
then the arrow for the cross product will end up pointing in the opposite direction.
There is a rule for using your right hand to remember
the direction in which the arrow representing the cross product is pointing.
If you point your right hand in the direction of the first arrow,
and then curl your fingers in the direction of the second arrow,
your thumb will end up pointing in the direction of the cross product.
Much more information is available in the other videos on this channel,
and please subscribe for notifications when new videos are ready.
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About this Video
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Dec 22, 2024 Last updated |
Video Description: Cross Product of Vectors for Civil Engineering (CE) 2024 is part of Engineering Mechanics preparation.
The notes and questions for Cross Product of Vectors have been prepared according to the Civil Engineering (CE) exam syllabus.
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