37. 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 concept of the cross product of two vectors, also known as the vector product. We are given two three-dimensional vectors, a⃗ and b⃗, and their cross product is expressed in terms of the unknown coefficients α, β, and γ. Our objective is to calculate the cross product, identify the values of these three coefficients, and then determine their sum.

The given vectors are:

a⃗ = 4î − 2ĵ + 6k̂

b⃗ = 7î + ĵ − 12k̂

The cross product is represented as:

a⃗ × b⃗ = αî + βĵ + γk̂

Therefore, after calculating a⃗ × b⃗, we can compare the coefficients of î, ĵ, and k̂ to obtain the values of α, β, and γ.

Writing the Components of the Given Vectors

Before calculating the cross product, it is useful to write the vectors in component form. The coefficients of î, ĵ, and k̂ represent the x, y, and z components of a vector.

For the first vector:

a⃗ = 4î − 2ĵ + 6k̂

Therefore, its components are:

a⃗ = (4, −2, 6)

Similarly, for the second vector:

b⃗ = 7î + ĵ − 12k̂

Therefore, its components are:

b⃗ = (7, 1, −12)

Formula for the Cross Product of Two Vectors

The cross product of two three-dimensional vectors can be calculated using the determinant method. If:

a⃗ = a₁î + a₂ĵ + a₃k̂

and

b⃗ = b₁î + b₂ĵ + b₃k̂

then their cross product is written as:

a⃗ × b⃗ = | î    ĵ    k̂ ; a₁    a₂    a₃ ; b₁    b₂    b₃ |

For the given vectors, the determinant becomes:

a⃗ × b⃗ = | î    ĵ    k̂ ; 4    −2    6 ; 7    1    −12 |

Expanding the Determinant to Find the Cross Product

Expanding the determinant along the first row gives:

a⃗ × b⃗ = î[(-2)(-12) − (6)(1)] − ĵ[(4)(-12) − (6)(7)] + k̂[(4)(1) − (-2)(7)]

We now calculate the coefficient of each unit vector separately.

Calculating the Coefficient of î

The coefficient of î is:

(−2)(−12) − (6)(1)

Multiplying the terms gives:

24 − 6 = 18

Therefore, the î-component of the cross product is:

18î

Calculating the Coefficient of ĵ

The coefficient of ĵ contains a negative sign because of the standard cofactor expansion of the determinant:

−[(4)(−12) − (6)(7)]

Calculating the products inside the brackets:

−[−48 − 42]

This becomes:

−[−90] = 90

Therefore, the ĵ-component of the cross product is:

90ĵ

Calculating the Coefficient of k̂

The coefficient of k̂ is:

(4)(1) − (−2)(7)

Therefore:

4 − (−14) = 4 + 14 = 18

Hence, the k̂-component of the cross product is:

18k̂

Writing the Complete Cross Product

Combining the calculated coefficients of î, ĵ, and k̂, we obtain:

a⃗ × b⃗ = 18î + 90ĵ + 18k̂

The question states that:

a⃗ × b⃗ = αî + βĵ + γk̂

Comparing the coefficients of the corresponding unit vectors, we get:

α = 18

β = 90

γ = 18

Calculating the Value of α + β + γ

Now, we add the values of α, β, and γ:

α + β + γ = 18 + 90 + 18

Therefore:

α + β + γ = 126

Verification Using the Direct Cross Product Formula

The result can also be verified using the direct component formula for the cross product. For two vectors 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 the values a⃗ = (4, −2, 6) and b⃗ = (7, 1, −12) gives:

a⃗ × b⃗ = [(-2)(-12) − (6)(1)]î + [(6)(7) − (4)(-12)]ĵ + [(4)(1) − (-2)(7)]k̂

After simplification:

a⃗ × b⃗ = 18î + 90ĵ + 18k̂

This confirms that:

α = 18, β = 90, and γ = 18

Final Answer

The required value of α + β + γ is:

126

Correct Answer: 126