Let a=2i^−3j^+k^, b=3i^+2j^+5k^ and a vector c be such that (a−c)×b=−18i^−3j^+12k^ and a⋅c=3. If b×c=d, then ∣a⋅d∣ is equal to:
Get the complete, step-by-step math solution for: "Let {a} = 2 {i} - 3 {j} + {k}, {b} = 3 {i} + 2 {j} + 5 {k} and a vector {c} be such that ( {a} - {c}) × {b} = -18 {i} - 3 {j} + 12 {k} and {a} · {c} =...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Expand the cross product
We begin by expanding the given cross product using the distributive property of the cross product. This allows us to separate the terms involving a and c.
Step 2: Substitute and rearrange the equation
We substitute the expanded form back into the given equation. We are also given that b×c=d. Since the cross product is anti-commutative, c×b=−(b×c)=−d. Substituting this into the equation simplifies it further.
Step 3: Calculate a×b
Now, we calculate the cross product of a and b using the determinant formula. This gives us a specific vector value for a×b.
Step 4: Solve for d
Substitute the calculated value of a×b into the equation from step 2. Then, rearrange the equation to solve for d.
Step 5: Calculate a⋅d
Finally, we calculate the dot product of a and d using their component forms. The dot product is the sum of the products of corresponding components.
Step 6: Find the magnitude of a⋅d
The problem asks for the magnitude of a⋅d, which is the absolute value of the scalar result obtained in the previous step.