Let a=i^+j^+k^, b=2i^+2j^+k^ and d=a×b. If c is a vector such that a⋅c=∣c∣, ∣c−2a∣2=8 and the angle between d and c is 4π, then ∣10−3b⋅c∣+∣d×c∣2 is equal to
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Step-by-Step Solution
Step 1: Calculate vectors a, b, and d and their magnitudes
First, we calculate the magnitudes of vectors a and b. Then, we compute the cross product d=a×b using the determinant formula. Finally, we find the magnitude of vector d. These values will be used in subsequent calculations.
Step 2: Determine the magnitude of vector c
We use the given condition ∣c−2a∣2=8. Expanding the dot product, we get ∣c∣2−4a⋅c+4∣a∣2=8. Substituting the given a⋅c=∣c∣ and the calculated ∣a∣=3, we form a quadratic equation in terms of ∣c∣. Solving this equation yields ∣c∣=2.
Step 3: Calculate b⋅c
We use the given conditions a⋅c=∣c∣ and the angle between d and c is 4π to find d⋅c. We express c as a linear combination of a, d, and a vector perpendicular to both, which is parallel to a×d. By substituting the known dot products and magnitudes, we solve for the scalar coefficient k. This yields two possible values for k, leading to two possible vectors for c and consequently two possible values for b⋅c.
Step 4: Calculate ∣d×c∣2
We use the formula for the magnitude of the cross product: ∣d×c∣=∣d∣∣c∣sinθ, where θ is the angle between d and c. We substitute the calculated magnitudes of d and c, and the given angle 4π, to find ∣d×c∣2.
Step 5: Calculate the final expression for both cases
We substitute the two possible values for b⋅c and the calculated value of ∣d×c∣2 into the given expression. In both cases, the result is 6.