Physics - Grade XI or Standard XI
Dot Product or Scalar Product of Two Vectors
Dot product or scalar product of two vectors:
The product of magnitudes of two non-zero vectors and cosine of the angle θ between them is called a dot product or scalar product of two vectors. It is given by the following formula (bold letters denote vectors):
Example 1 of dot product of two vectors:
Let P = 3 i - 4 j + 7 k and let Q = -5 i + 9 j + 6 k. Find the dot product of P and Q.
Solution:
If P = Px i + Py j + Pz k and Q = Qx i + Qy j + Qz k, then dot product of P and Q is given by:
P • Q = (Px × Qx) + (Py × Qy) + (Pz × Qz)
The values of components of P and Q are as follows:
Px = 3, Py = -4, Pz = 7, Qx = -5, Qy = 9, and Qz = 6.
Put these values in the above given formula and we get:
P • Q = (Px × Qx) + (Py × Qy) + (Pz × Qz)
P • Q = (3 × -5) + (-4 × 9) + (7 × 6) = -9
P • Q = -9 ----------------------(Answer)
Example 2 of dot product of two vectors:
Magnitude of vector P is 25 units, magnitude of vector Q is 30 units, and angle θ between them is 60o. Find the dot product of vectors P and Q.
Solution:
Dot product of two vectors P and Q can be computed using the formula:
P • Q = P × Q × cosθ
Put the values in this formula and we get:
P • Q = P × Q × cosθ = 25 units × 30 units × cos(60o)
P • Q = 25 units × 30 units × 0.5000 = 375
P • Q = 375 -------------------------(Answer)
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