In physics, vectors can be multiplied either by a scalar or by another vector. One important type of vector multiplication is the scalar product, also known as the dot product. The result of a scalar product is a scalar quantity.
The scalar product is widely used in physics to define quantities such as work, energy, and power. For example, the work done by a force is defined as the scalar product of the force vector and the displacement vector.
The scalar product of two vectors A and B is defined as:
\boxed{\overrightarrow{\rm A} \cdot \overrightarrow{\rm B}= |\overrightarrow{\rm A}|\, | \overrightarrow{\rm B} | \cos\theta} where
|\mathbf{A}| \, \, and \, \, |\mathbf{B}| are the magnitudes of the vectors, and\theta is the angle between them.Since the product is represented by a dot (·), it is called the dot product.
Scalar Product in terms of Unit Vector Representation
In the unit vector representation of vectors, i, j, and k are along the x-axis, y-axis, and z-axis, respectively. The scalar product can be calculated as
where,
Matrix Representation of Scalar Products
It is useful to represent vectors as row or column matrices instead of as the above unit vectors. If we treat vectors as column matrices of their x, y, and z components, then the transposes of these vectors would be row matrices.
Now therefore, the Matrix A and Matrix B look like this:
The matrix product of these 2 matrices will give us the scalar product of the 2 matrices, which is the sum of corresponding components of the given 2 vectors; the resulting number will be the scalar product of vector A and vector B.
Physical Interpretation
Geometrically, the scalar product represents the product of the magnitude of one vector and the component of the other vector along its direction. It measures how much one vector acts in the direction of another.
Geometrical interpretation
The product of two nonzero vectors can be visualized as multiplying the magnitude of any one of the vectors by the magnitude of the projection of the other vector upon it.
Case 1: When the angle between the two vectors is 0° < θ < 90°, then the scalar product is positive.
Case 2: When the angle between the two vectors is 90° < θ < 180°, then the scalar product is negative.
Case 3: When the angle between the two vectors is θ = 90°, then the scalar product is 0 (zero).
Special Cases of Scalar Product
(1) Scalar product of two parallel vectors: The scalar product of two parallel vectors is simply the product of the magnitudes of the two vectors. As the angle between vectors when they are parallel is 0 degrees and cos 0 = 1.
Therefore,
a.b = |a| × |b| cos 0
= |a| × |b|
(2) Scalar Product of Two Antiparallel Vectors: The scalar product of two antiparallel vectors is the negative of the product of the magnitudes of the two vectors.
a.b cos 180 = −|a| |b| (Since, cos 180 = -1)
(3) Scalar product of two orthogonal vectors: The scalar product of two orthogonal vectors is 0.
a.b cos 90 = 0 (Since cos 90 = 0)
Properties of Scalar Product (Dot Product)
1. Commutative Properties of Scalar Multiplication: If a and b are two vectors, then:
a.b = b.a
As, a.b = |a||b| cos θ and b.a = |b||a| cos θ Therefore a.b = b.a
2. Distributive Over Vector Addition: If a, b, and c are vectors, then
a.(b+c) = a.b + a.c
3. Scalar Multiplication: The scalar product is consistent with scalar multiplication. If u and v are constants and a and b are vectors, then,
(u a). (v b) = u v (a. b)
4. Orthogonality: Two vectors are orthogonal if their scalar product is zero.
i.e., vectors u and v are orthogonal if u . v = 0.
Applications of Scalar Product
The scalar product, also known as the dot product, serves multiple purposes in vector theory, including:
Projection of a Vector: The scalar product is essential for computing the projection of one vector onto another. Specifically, the projection of vector a onto vector b can be expressed as (a·b)/|b|, while the projection of vector b onto vector a is (a·b)/|a|.
Scalar Triple Product: The scalar product also plays a role in determining the scalar triple product of three vectors. The formula for this is a·(b × c) = b·(c × a) = c·(a × b), illustrating its cyclic nature.
Angle Between Two Vectors: The scalar product is used to calculate the angle between two vectors. This is done using the cosine formula: cos(θ) = (a·b)/(|a||b|), which relates the dot product of the vectors to the product of their magnitudes and the cosine of the angle between them.
Solved Problems
Problem 1: Find the scalar product of vector A = 2i + 5j + 3k and vector B = 3i + j + 2k.
A. B = (2 * 3) + (5 * 1) +(3 * 2)
= 6 + 5 + 6
= 17
Problem 2: A particle covers a displacement from position 2i + j + 2k to 3i + 2j + 5k due to a uniform force of (7i + 5j + 2k) N. If the displacement is in meters, calculate the work done.
Given,
Final position P2 = 3i + 2j +5k
Initial position P1 = 2i +j +2k
Force F = ( 7 i + 5 j + 2 k ) N
So
The displacement D = P2 – P1
D = ( 3 – 2 ) i + ( 2 -1 ) j + ( 5 – 2 ) k
= i + j + 3k
Work done = F . D
= ( 7 i + 5 j + 2 k ) . ( i + j + 3k )
= ( 7*1 ) + ( 5*1 ) + ( 2*3 )
= 7 + 5 + 6
= 18 J
Problem 3: Find the value of m such that vectors A = 2i + 3j + k and B = 3i + 2j + mk may be perpendicular.
Given,
A and B are perpendicular
so A . B = 0
( 2 i + 3j + k ) . ( 3i + 2j + mk ) = 0
( 2 * 3 ) + ( 3 * 2 ) + ( 1 * m ) = 0
6 + 6 + m = 0
12 + m = 0
m = -12
Problem 4: Prove that vectors U = 2i + 3j + k and V = 4i - 2j - 2k are perpendicular to each other.
Given,
U = 2i + 3j + k
V = 4i - 2j - 2k
to prove U and V are perpendicular to each other
U . V = ( 2i + 3j + k ) . ( 4i - 2j - 2k )
= ( 2 * 4 ) + ( 3 * -2 ) + ( 1 * -2 )
= 8 - 6 - 2
= 0
we know that if scalar product of two vectors is 0 then they are perpendicular to each other.
U . V = 0 so, U and V are perpendicular to each other
Hence proved
Problem 5: Check whether the vectors A = 4ax - 2ay - az and B = ax + 4ay - az are perpendicular to each other.
A⋅B = (4ax−2ay−az)⋅(ax+4ay−az)
= 4(1) + (-2)(4) + (-1)(-1)
= 4 - 8 + 1
= −3
Using the Dot Product Relation
A.B=|A|.|B| cosθ
Since
A . B= −3 ≠ 0
cosθ≠0
Therefore,
θ≠90∘
Hence, vectors A & B are not perpendicular to each other.
Unsolved Problems
Question 1: Two vectors are perpendicular to each other. Their magnitudes are 4 and 7. Find their dot product.
Question 2: Find A.B (dot product), given A = 2i−3j+4k and B = i+2j−2k.
Question 3: A force F = 3i + 4j acts on a body and causes displacement, s = 5i + 2j. Find the work done.
Question 4: Find the projection of vector A = 6i+8j on vector B = 3i+4j