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Chapter 11: Problem 52

Find the value of \(a\) such that \(\langle a, a, 2\rangle \times\langle 1, a, 3\rangle=\langle 2,-4,2\rangle\)

Short Answer

Expert verified
Answer: There is no value of \(a\) that satisfies the given condition.

Step by step solution

01

Calculate the cross product

To find the cross product of the two given vectors, we can use the formula \((\vec{u} \times \vec{v}) = (u_2v_3 - u_3v_2, -(u_1v_3 - u_3v_1), u_1v_2 - u_2v_1)\). Let \(\vec{u} = \langle a, a, 2\rangle\) and \(\vec{v} = \langle 1, a, 3\rangle\). Applying the formula, we have: \((\vec{u} \times \vec{v}) = ((a)(3) - (2)(a), -((a)(3) - (2)(1)), (a)(a) - (a)(1))\) Simplify to get: \((\vec{u} \times \vec{v}) = (a - 2, -a + 2, a^2 - a)\)
02

Compare the cross product to the given result

We are given that the cross product should equal \(\langle 2, -4, 2\rangle\). Now, we compare the components of our calculated cross product to the given result: \(a - 2 = 2\) (component 1) \(-a + 2 = -4\) (component 2) \(a^2 - a = 2\) (component 3)
03

Solve the equations to get the value of \(a\)

We shall first solve the equation for component 1 to find the value of \(a\): \(a - 2 = 2\) \(a = 4\) Now, let's check if this value of \(a\) satisfies the equations for components 2 and 3.
04

Check if the calculated value of \(a\) holds for the other components

Substituting \(a=4\) in the equation for component 2: \(-4 + 2 = -4\) \(-2 = -4\) This does not hold true, so we may have another value for \(a\). Now, let's examine the quadratic equation of component 3: \(a^2 - a = 2\) \(a^2 - a - 2 = 0\) Factoring this quadratic equation, \((a-2)(a+1) =0\) \(a = 2\) or \(a = -1\)
05

Verify with the other components

Now, let's see if \(a=2\) or \(a=-1\) works for the other components. For \(a=2\): Component 1: \(2 - 2 = 2\) (false) Component 2: \(-2 + 2 = -4\) (false) For \(a=-1\): Component 1: \((-1) - 2 = 2\) (false) Component 2: \(1 + 2 = -4\) (false)
06

Conclusion

Since none of the possible values for \(a\) satisfy all the components, there is no value of \(a\) that satisfies the given condition.

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Most popular questions from this chapter

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