- Vector multiplied with scalar gives a resultant of vector.
- Vector multiplied with vector gives a resultant of scalar(Dot Product)
- Vector multiplied with vector gives a resultant of vector(Cross Product)
Here is the explanation in detail for each time of multiplication.
CASE ONE :
1. When a vector is multiplied by a scalar its products is a vector whose magnitude is equal to the scalar times the magnitude of the given vector.
2.The direction of a vector is same as the given vector, if the scalar is positive and opposite if the scalar is negative.
Example :
P = m v where P is momentum, m is mass and v is velocity.
F = m a where F is force , m is mass and a is acceleration.
NOTE :
A vector multipllied by another vector may give a scalar (or) a vector. Hence there are two types of products for multiplication of two types of products for multiplication of two vectors.
a) dot product (or) Scalar product
b) cross product (or) vector product
CASE TWO SCALAR PRODUCT (OR) DOT PRODUCT PROPERTIES :
1 . When the magnitude of one vector along another vector is effective then the dot product of two vectors is taken.
2. The dot product of two vectors is a scalar.
3. The scalar product of two vectors and is a.b = ab cos θ
4. Scalar product is commutative i.e. a.b = b.a
5 . Scalar product is distributive i.e a.(b+c) = a.b + a.c
6 . The scalar product of two parallel vectors is maximum I.e when θ = 0
7 . The scalar product of two opposite vectors is negative i.e when θ = 180.
8 . The scalar product of two perpendicular vectors is zero when θ = 90.
9 . In case of unit vectors i.i = j.j = k.k = 1 i.e i.i = 1*1*cos 0 = 1
10 . Similarly i.j = j.k = k.i = 0 since i.j = 1 *1* cos 90 = 0.
11 . In terms of Components A.B = AxBx + AyxBy + AzBz .
APPLICATIONS OF DOT PRODUCT :
1 . W = F.S Dot product of force and displacement is work .
2 . P = F.V Dot product of force and velocity is power.
3 . E = mgh Dot product of gravitational force and vertical displacement is P.E.
4 . Ø = B.A Dot product of area vector and magnetic flux density vector.
5. Angle between the two vectors a and b is a.b/|a| |b| .
RELATED POST
BASICS OF VECTORS PART ONE AND TWO.
CASE ONE :
1. When a vector is multiplied by a scalar its products is a vector whose magnitude is equal to the scalar times the magnitude of the given vector.
2.The direction of a vector is same as the given vector, if the scalar is positive and opposite if the scalar is negative.
Example :
P = m v where P is momentum, m is mass and v is velocity.
F = m a where F is force , m is mass and a is acceleration.
NOTE :
A vector multipllied by another vector may give a scalar (or) a vector. Hence there are two types of products for multiplication of two types of products for multiplication of two vectors.
a) dot product (or) Scalar product
b) cross product (or) vector product
CASE TWO SCALAR PRODUCT (OR) DOT PRODUCT PROPERTIES :
1 . When the magnitude of one vector along another vector is effective then the dot product of two vectors is taken.
2. The dot product of two vectors is a scalar.
3. The scalar product of two vectors and is a.b = ab cos θ
4. Scalar product is commutative i.e. a.b = b.a
5 . Scalar product is distributive i.e a.(b+c) = a.b + a.c
6 . The scalar product of two parallel vectors is maximum I.e when θ = 0
7 . The scalar product of two opposite vectors is negative i.e when θ = 180.
8 . The scalar product of two perpendicular vectors is zero when θ = 90.
9 . In case of unit vectors i.i = j.j = k.k = 1 i.e i.i = 1*1*cos 0 = 1
10 . Similarly i.j = j.k = k.i = 0 since i.j = 1 *1* cos 90 = 0.
11 . In terms of Components A.B = AxBx + AyxBy + AzBz .
APPLICATIONS OF DOT PRODUCT :
1 . W = F.S Dot product of force and displacement is work .
2 . P = F.V Dot product of force and velocity is power.
3 . E = mgh Dot product of gravitational force and vertical displacement is P.E.
4 . Ø = B.A Dot product of area vector and magnetic flux density vector.
5. Angle between the two vectors a and b is a.b/|a| |b| .
RELATED POST
BASICS OF VECTORS PART ONE AND TWO.
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