JEE > Mathematics (Maths) Class 12 > Overview of Scalar and Vector Product of 2 Vectors
Overview of Scalar and Vector Product of 2 Vectors Video Lecture - Mathematics (Maths) Class 12 - JEE
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FAQs on Overview of Scalar and Vector Product of 2 Vectors Video Lecture - Mathematics (Maths) Class 12 - JEE
| 1. What is the scalar product of two vectors? |  |
The scalar product, also known as the dot product, is a mathematical operation that yields a scalar quantity. It is calculated by multiplying the magnitudes of two vectors and the cosine of the angle between them. The formula for the scalar product of two vectors A and B is A · B = |A| |B| cos(θ), where θ is the angle between the vectors.
| 2. What is the vector product of two vectors? | |
The vector product, also known as the cross product, is a mathematical operation that yields a vector quantity. It is calculated by taking the cross product of the magnitudes of two vectors and the sine of the angle between them. The formula for the vector product of two vectors A and B is A × B = |A| |B| sin(θ) n, where θ is the angle between the vectors and n is a unit vector perpendicular to both A and B.
| 3. How is the scalar product related to the angle between two vectors? | |
The scalar product of two vectors is directly related to the angle between them. The scalar product is equal to the product of the magnitudes of the vectors and the cosine of the angle between them. If the scalar product is positive, the angle between the vectors is acute. If the scalar product is negative, the angle between the vectors is obtuse. And if the scalar product is zero, the vectors are perpendicular to each other.
| 4. What does the vector product of two vectors represent? | |
The vector product of two vectors represents a new vector that is perpendicular to both of the original vectors. This new vector's magnitude is equal to the product of the magnitudes of the original vectors and the sine of the angle between them. The direction of the vector product can be determined using the right-hand rule, where the thumb points in the direction of the vector product when the fingers curl from the first vector to the second vector.
| 5. In what situations are the scalar and vector products used? | |
The scalar product is commonly used in physics and engineering to calculate work done, find projections of one vector onto another, determine angles between vectors, and solve problems involving forces and displacements. On the other hand, the vector product is frequently used to calculate torque, determine the area of a parallelogram formed by two vectors, find the direction of magnetic fields, and solve problems involving angular momentum and rotational motion.
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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