Multiplication of a Vector by a Scalar

In mathematics and physics, the vector is a quantity that has direction and magnitude. There are several different types of vectors. These are zero vector, unit vector, coinitial vectors, equal vectors, and negative of a vector. In physics and mathematics, scalar has only magnitude and no direction. 

The result of multiplying a vector with a scalar is determined by multiplying the similar components of a vector and a scalar, which is magnitude. The product of the two magnitudes will result in a new vector with the same direction. 

Multiplication of a vector by a scalar 

• If λ is a scalar and λ a vector, their product will be denoted as  a. 
• According to the value of the (either positive or negative), the new vector a will have the same direction as vector a or opposite. 
• The magnitude of the new vector λ a is the magnitude of scalar times the magnitude of the vector.lλ al =lλl al
• If λ= 1/lal, then lλal = lλlal = 1/lalal = 1

However, provided that vector a 0. Therefore, a shouldn’t be a null vector for the above condition to hold. 

Components of a Vector 

• Let vector OP be represented as zi + yj + zk, then

• x, y, and z are the scalar components of the vector, and the vector mentioned above is called its component form. x, y, and z are also known as rectangular components. 
• xi, yj, and zk are known as the vector components of the vector. 

• Let the component form of vectors a and b be a1i + a2j + a3k and b1i + b2j + b3k, respectively, then 

1. The resultant sum of the two vectors will be denoted as 

a + b = (a1 + b1)i + (a2 + b2)j + (a3 + c3)k

1. The resultant difference of the two vectors will be denoted as

a – b = (a1 – b1)i + (a2 – b2)j + (a3 – c3)k

1. If a1 = b1, a2 = b2, and a3 = b3, then the vectors a and b will be considered equal. 
2. Let λ be any scalar. Then the multiplication of vector a with will be represented as- 

λa = (λa1)i + (λa2)j + (λa3)k

Distributive laws for addition and multiplication of a vector by a scalar- 

• Let k and m be any scalars and a and b be any two vectors.

1. ka + ma = (k + m) a 
2. k(ma) = (km)a 
3. k (a + b) = ka + kb 

• For example, vector a = i – 2j with magnitude 7 units. 

The unit vector in the direction of vector a =  1/a *a = 1/√5 (i – 2j) = 1/√5*i – 2/√5*j 

The new vector of magnitude 7, which is in the direction of vector a is 

7 a = 7 (1/√5*i – 2/√5*j) = 7/√5*i – 14/√5*j

Vector Joining two points 

• The vector joining the two points can be calculated by – (x2 – x1)i + (y2 – y1)j + (z2 – z1)k. 
• The magnitude of a vector joining two points is = √(x2−x1)2+(y2−y1)2+(z2−z1)2 
• For example, there are two vectors, P (2,3,0) and Q (-1, -2, -3). There is a vector directed from P to Q that joins the two vectors together. 

Therefore, the vector joining the two points is = (-1 – 2)i + (- 2 – 3)j + (- 3 – 0)k  

= -3i – 5j – 4k. 

How to multiply the vector by a scalar? 

• To multiply the vector by a scalar is very easy; simply multiply the magnitudes of the vector with the scalar. 
• The resultant new vector will have the same direction. 
• For example, let c = ⟨8⟩. Find 7c. 

7c = 7⟨8⟩

      = ⟨7(8)⟩

      = ⟨56⟩

 Scalar multiple of a vector 

• When a vector is multiplied by a scalar, the magnitude of the vector changes. 
• The direction of the scalar multiple remains the same as the input vector.
• Either the scalar is zero or the vector is zero if the scalar multiple of a vector is zero. 

What are the scalar and vector products of two vectors? 

• The scalar and vector products of two vectors give you a scalar value and a vector value, respectively. 
• The scalar product of two vectors is also known as the dot product. 
• The vector product of two vectors is also known as the cross product. 

• For example, let’s find the scalar and vector products of a = (3i – 4j + 5k) and b = (-2i + j – 3k). 

a × b = (3i – 4j + 5k) × (-2i + j – 3k) 

= – 6 – 4 – 15 

= – 25

a × b = li     j    k  3-4 5  -2 1-3 l 

a × b = 7i – j – 5k 

Conclusion

The vector is a quantity that is referred to have both direction as well as magnitude. Scalar is the quantity that consists of only the magnitude when both the vector and scalar quantity are multiplied, we get to know the magnitude of both vector and scalar as a product and the single direction that we attain through the vector quantity. The scalar quantity is denoted as while the vector is denoted with a the product of both will be a. The product can either be positive or negative. The vector and scalar quantities are easy to calculate and decipher.