Class 12 Exam > Class 12 Videos > Concepts: Scalar , Vector Product - Vector Algebra
Concepts: Scalar , Vector Product - Vector Algebra Video Lecture - Class 12
FAQs on Concepts: Scalar , Vector Product - Vector Algebra Video Lecture - Class 12
| 1. What is the difference between scalar and vector product in vector algebra? |  |
Ans. The scalar product, also known as the dot product, is a binary operation that takes two vectors and returns a scalar quantity. It is calculated by multiplying the magnitudes of the vectors and the cosine of the angle between them. On the other hand, the vector product, also known as the cross product, is a binary operation that takes two vectors and returns a vector quantity. It is calculated by multiplying the magnitudes of the vectors, the sine of the angle between them, and the unit normal to the plane containing the vectors.
| 2. How is the scalar product useful in physics and engineering? | |
Ans. The scalar product has various applications in physics and engineering. It is used to calculate work done, determine the angle between two vectors, find the projection of one vector onto another, calculate the magnitude of a vector, and determine whether two vectors are perpendicular or parallel.
| 3. What are the properties of the vector product in vector algebra? | |
Ans. The vector product has several properties, including:
- Distributive property: a × (b + c) = (a × b) + (a × c)
- Anti-commutative property: a × b = - (b × a)
- Scalar multiplication property: (k*a) × b = k*(a × b) = a × (k*b) where k is a scalar
- Triple scalar product property: a × (b × c) = (a · c) * b - (a · b) * c
| 4. How can the scalar and vector products be used to find the area of a parallelogram? | |
Ans. The magnitude of the vector product of two vectors is equal to the area of the parallelogram formed by those vectors. So, to find the area of a parallelogram, we can take the magnitude of the vector product of two adjacent sides of the parallelogram.
| 5. What is the geometric interpretation of the scalar and vector products? | |
Ans. The scalar product can be interpreted as the product of the magnitudes of two vectors and the cosine of the angle between them. It gives us information about the degree of alignment or opposition between the vectors. The vector product, on the other hand, can be interpreted as a vector that is perpendicular to both of the original vectors. Its direction follows the right-hand rule, and its magnitude is equal to the area of the parallelogram formed by the vectors.
Text Transcript from Video
hello friends this video on vector
algebra part 16 is brought you by exam
feel calm no more free from exam before
watching this video please make sure
that you have watched part 1 part 15 now
let's understand a very critical topic
called product of two vectors till now
we have done some additional vectors we
also multiply some scalar into vector so
till now what we have done with the
added vector we subtracted that vector
right he multiplied multiplied scalar to
inter scalar to a vector that is some
number like three four five seven it
multiplied to it we have done this Stila
but we never multiply a vector to it
first let's understand why do we need
that so to understand this we should
know a little bit of physics because of
two examples which I am going to cover
has physics in it I could not get the
better example if you have let me know
will make videos on that so let's
suppose there is a good'n stick wooden
stick and you want to move this wooden
stick what you will do you will apply
some force on this correct if you apply
force on this what will happen is
gradually this will move correct so this
guy move from here to here why because
you apply the force numbers in the whole
thing you found that some work was done
and the work was done was nothing but
force into the system here if you see my
force is a vector into displacement
actually and my displacement is also a
vector
right so I multiply two vectors and I
know that the work done which I got is
scalar so we have force we have
displacement right we multiply these two
and we got work done that is a scalar
quantity let's take a sample we have
similar wooden log this time we fixed it
with a nail this is the nail and we
fixed it on the wall now again let's
apply force now what will happen this
will rotate if you see it didn't move it
totally correct so in this case also you
see we had this is let's suppose the
vector R vector that is distance vector
from center I had this force
I had this our victim a multiplied these
two I got something called dot here my
force is a vector minus R is a vector
and what I get is also a vector correct
so if you see when I multiply vector in
one case I got scalar as output in
another case I got a vector as of that
means I need to kind of vector
multiplication this is the need for two
kind of vectors multiplications so if I
have a vector one and make the V so that
means I need to kind of vector
multiplication and that's what we call
dot product and cross Poland so this is
cross product and it is dot product in
case of cross product you get vector as
output in case of dot product we get
scalar as out so we'll learn more about
these but let this background we're in
first case the force was
line there is a displacement both force
and displacement is a vector but what we
talk about is the work and work is
nothing but force into displacement and
then scalar so we have two vectors we
did a dot product and we got a scalar in
second case we had this force we had
this R vector we applied force and
listens R from this point and it
produced a toc-toc it's in the vector so
we are looking for cross product here
then I am saying that force cross R
vector you get at all that is also a
vector
thank you visit exam viacom to watch
free education videos try free online
tests get the best quality study
materials study from the best tutors and
mentors and much more thanks and screen
algebra part 16 is brought you by exam
feel calm no more free from exam before
watching this video please make sure
that you have watched part 1 part 15 now
let's understand a very critical topic
called product of two vectors till now
we have done some additional vectors we
also multiply some scalar into vector so
till now what we have done with the
added vector we subtracted that vector
right he multiplied multiplied scalar to
inter scalar to a vector that is some
number like three four five seven it
multiplied to it we have done this Stila
but we never multiply a vector to it
first let's understand why do we need
that so to understand this we should
know a little bit of physics because of
two examples which I am going to cover
has physics in it I could not get the
better example if you have let me know
will make videos on that so let's
suppose there is a good'n stick wooden
stick and you want to move this wooden
stick what you will do you will apply
some force on this correct if you apply
force on this what will happen is
gradually this will move correct so this
guy move from here to here why because
you apply the force numbers in the whole
thing you found that some work was done
and the work was done was nothing but
force into the system here if you see my
force is a vector into displacement
actually and my displacement is also a
vector
right so I multiply two vectors and I
know that the work done which I got is
scalar so we have force we have
displacement right we multiply these two
and we got work done that is a scalar
quantity let's take a sample we have
similar wooden log this time we fixed it
with a nail this is the nail and we
fixed it on the wall now again let's
apply force now what will happen this
will rotate if you see it didn't move it
totally correct so in this case also you
see we had this is let's suppose the
vector R vector that is distance vector
from center I had this force
I had this our victim a multiplied these
two I got something called dot here my
force is a vector minus R is a vector
and what I get is also a vector correct
so if you see when I multiply vector in
one case I got scalar as output in
another case I got a vector as of that
means I need to kind of vector
multiplication this is the need for two
kind of vectors multiplications so if I
have a vector one and make the V so that
means I need to kind of vector
multiplication and that's what we call
dot product and cross Poland so this is
cross product and it is dot product in
case of cross product you get vector as
output in case of dot product we get
scalar as out so we'll learn more about
these but let this background we're in
first case the force was
line there is a displacement both force
and displacement is a vector but what we
talk about is the work and work is
nothing but force into displacement and
then scalar so we have two vectors we
did a dot product and we got a scalar in
second case we had this force we had
this R vector we applied force and
listens R from this point and it
produced a toc-toc it's in the vector so
we are looking for cross product here
then I am saying that force cross R
vector you get at all that is also a
vector
thank you visit exam viacom to watch
free education videos try free online
tests get the best quality study
materials study from the best tutors and
mentors and much more thanks and screen
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Jan 06, 2025
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