Multiplication of Vectors - Scalar Product(Dot Product)

Imagine you're pushing a heavy box across the floor. If you push directly forward, all your force helps move the box. But if you push at an angle, only part of your force actually moves the box forward - this is exactly what the scalar product measures!

The scalar product or dot product of two nonzero vectors \[\vec P\] and \[\vec Q\] is defined as the product of the magnitudes of the two vectors and the cosine of the angle θ between the two vectors.

The scalar product is defined when vectors are given in their component form.

Fig. 2.9: Projection of vectors.

Step 1: Define the vectors
Let two vectors P and Q be:
P = P_x\[\hat i\] + P_y\[\hat j\] + P_z\[\hat k\]
Q = Q_x\[\hat i\] + Q_y\[\hat j\] + Q_z\[\hat k\]

Step 2: Write the dot product
P ⋅ Q = (P_x\[\hat i\] + P_y\[\hat j\] + P_z\[\hat k\]) ⋅ (Q_x\[\hat i\] + Q_y\[\hat j\] + Q_z\[\hat k\])

Step 3: Distribute the terms
Multiply each component of P by each component of Q:
= (P_xQ_x)(\[\hat i\] . \[\hat i\]) + (P_xQ_y)(\[\hat i\] . \[\hat j\]) + (P_xQ_z)(\[\hat i\] . \[\hat k\])
+ (P_yQ_x)(\[\hat j\] . \[\hat i\]) + (P_yQ_y)(\[\hat j\] . \[\hat j\]) + (P_yQ_z)(\[\hat j\] . \[\hat k\])
+ (P_zQ_x​)(\[\hat k\] . \[\hat i\]) + (P_zQ_y)(\[\hat k\] . \[\hat j\]) + (P_zQ_z)(\[\hat k\] . \[\hat k\])

Step 4: Simplify using properties of unit vectors
We know that the dot product of a unit vector with itself is 1 (\[\hat i\] . \[\hat i\] = 1), and the dot product with any other perpendicular unit vector is 0 (\[\hat i\] . \[\hat j\] = 0).
This simplifies the expression to:
= P_xQ_x(1) + 0 + 0
+ 0 + P_yQ_y(1) + 0
+ 0 + 0 + P_zQ_z (1)

Step 5: Final Formula
The final result is the sum of the products of the corresponding components:
P ⋅ Q = P_xQ_x + P_yQ_y + P_zQ_z

• The scalar product is very useful in physics.
• It makes mathematical formulas and their derivations more elegant and simple.
• It is used to define and calculate the Work Done by a force
  (W = \[\vec F\] ⋅ \[\vec S\]).
• It is used to calculate Power, which is the rate of doing work (P = \[\vec F\] ⋅ \[\vec v\]).

Find the scalar product: \[\vec v_1\] = \[\hat i\] + 2\[\hat j\] + 3\[\hat k\] and \[\vec v_2\] = 3\[\hat i\] + 4\[\hat j\] − 5\[\hat k\]

• Step 1: Identify components
  \[\vec v_1\] : P_x =1, P_y =2, P_z = 3
  \[\vec v_2\] : Q_x = 3, Q_y = 4, Q_z = −5
• Step 2: Apply the formula
  \[\vec v_1\] ⋅ \[\vec v_2\] = (1)(3) + (2)(4) + (3)(−5)
• Step 3: Calculate
  = 3 + 8 + (−15) = −4

Result: The scalar product is -4.

• Calculating Work Done: When you push a shopping cart, the work you do depends on the force you apply and the distance the cart moves. If you push at an angle, the dot product helps calculate the effective work done in the direction of motion. W = \[\vec F\] ⋅ \[\vec d\].
• Calculating Power: The power delivered by a car's engine can be described as the dot product of the force exerted by the engine and the velocity of the car. P = \[\vec F\] ⋅ \[\vec v\].
• Computer Graphics: In video games and 3D modeling, the dot product is used to determine how light reflects off a surface. It helps calculate the angle between a light source and a surface to create realistic shading and textures.