48. Let a⃗ = 4î − 2ĵ + 6k̂ and b⃗ = 7î + ĵ − 12k̂. If:
a⃗ × b⃗ = αî + βĵ + γk̂
then the value of α + β + γ equals _______.
Find α + β + γ Using the Cross Product of Two Vectors
Understanding the Given Vector Cross Product Problem
This question is based on the cross product of two vectors, also known as the vector product. Two three-dimensional vectors are given, and their cross product is expressed in terms of the unknown coefficients α, β, and γ. We need to calculate the cross product, compare the coefficients of î, ĵ, and k̂, and finally find the value of α + β + γ.
The given vectors are:
a⃗ = 4î − 2ĵ + 6k̂
and:
b⃗ = 7î + ĵ − 12k̂
The cross product is given in the form:
a⃗ × b⃗ = αî + βĵ + γk̂
Therefore, after calculating a⃗ × b⃗, we can directly identify the values of α, β, and γ.
Writing the Given Vectors in Component Form
A three-dimensional vector written in terms of the unit vectors î, ĵ, and k̂ can also be represented by its components.
For the first vector:
a⃗ = 4î − 2ĵ + 6k̂
the components are:
a⃗ = (4, −2, 6)
For the second vector:
b⃗ = 7î + ĵ − 12k̂
the components are:
b⃗ = (7, 1, −12)
We now use these components to calculate the required cross product.
Formula for the Cross Product of Two Vectors
If two vectors are written as:
a⃗ = a1î + a2ĵ + a3k̂
and:
b⃗ = b1î + b2ĵ + b3k̂
then their cross product can be calculated using the determinant:
a⃗ × b⃗ = | î ĵ k̂ ; a1 a2 a3 ; b1 b2 b3 |
Substituting the components of the given vectors:
a⃗ × b⃗ = | î ĵ k̂ ; 4 −2 6 ; 7 1 −12 |
We now expand this determinant along the first row.
Expanding the Determinant
Expanding the determinant gives:
a⃗ × b⃗ = î[(-2)(-12) − (6)(1)] − ĵ[(4)(-12) − (6)(7)] + k̂[(4)(1) − (-2)(7)]
Each component can now be simplified separately.
Calculating the î-Component
The coefficient of î is:
(−2)(−12) − (6)(1)
Multiplying the terms:
24 − 6
Therefore:
24 − 6 = 18
Hence, the î-component of the cross product is:
18î
Therefore:
α = 18
Calculating the ĵ-Component
The coefficient of ĵ is:
−[(4)(−12) − (6)(7)]
Calculating the products inside the brackets:
−[−48 − 42]
Therefore:
−[−90] = 90
Hence, the ĵ-component is:
90ĵ
Therefore:
β = 90
Calculating the k̂-Component
The coefficient of k̂ is:
(4)(1) − (−2)(7)
Simplifying:
4 − (−14)
Therefore:
4 + 14 = 18
Hence, the k̂-component is:
18k̂
Therefore:
γ = 18
Writing the Complete Cross Product
Combining all three components, we obtain:
a⃗ × b⃗ = 18î + 90ĵ + 18k̂
The question states that:
a⃗ × b⃗ = αî + βĵ + γk̂
Comparing the coefficients of the corresponding unit vectors gives:
α = 18
β = 90
γ = 18
Calculating the Value of α + β + γ
We now add the three values:
α + β + γ = 18 + 90 + 18
Adding the first two terms:
18 + 90 = 108
Therefore:
α + β + γ = 108 + 18
Hence:
α + β + γ = 126
Alternative Solution Using the Direct Cross Product Formula
The cross product can also be calculated directly using the component formula. For:
a⃗ = (a1, a2, a3)
and:
b⃗ = (b1, b2, b3)
the cross product is:
a⃗ × b⃗ = (a2b3 − a3b2)î + (a3b1 − a1b3)ĵ + (a1b2 − a2b1)k̂
Substituting:
a⃗ = (4, −2, 6)
and:
b⃗ = (7, 1, −12)
we get:
a⃗ × b⃗ = [(-2)(-12) − (6)(1)]î + [(6)(7) − (4)(-12)]ĵ + [(4)(1) − (-2)(7)]k̂
Simplifying:
a⃗ × b⃗ = (24 − 6)î + (42 + 48)ĵ + (4 + 14)k̂
Therefore:
a⃗ × b⃗ = 18î + 90ĵ + 18k̂
This again confirms:
α = 18, β = 90, and γ = 18
Final Answer
The cross product of the given vectors is:
a⃗ × b⃗ = 18î + 90ĵ + 18k̂
Therefore:
α = 18
β = 90
γ = 18
Hence:
α + β + γ = 126
Correct Answer: 126
