Let a=i^+2j^+3k^, b=3i^+j^−k^ and c be three vectors such that c is coplanar with a and b. If the vector c is perpendicular to b and a⋅c=5, then ∣c∣ is equal to
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Step-by-Step Solution
Step 1: Express vector c in terms of a and b
Since vector c is coplanar with vectors a and b, it can be expressed as a linear combination of a and b. Here, x and y are scalar constants that we need to determine.
Step 2: Use the perpendicularity condition
We are given that c is perpendicular to b. This means their dot product is zero. We substitute the expression for c and expand the dot product.
Step 3: Calculate dot products and solve for x and y
First, we calculate the dot product of a and b, and the dot product of b with itself (which is the square of its magnitude). Then, we substitute these values into the equation from the previous step to find a relationship between x and y.
Step 4: Use the given dot product condition
We are given that a⋅c=5. We substitute the expression for c again and expand the dot product.
Step 5: Calculate remaining dot product and solve for y
We calculate the dot product of a with itself. Then, we substitute this value and the relationship x=−211y into the equation from the previous step to solve for y.
Step 6: Find x and vector c
Now that we have y, we can find x using the relationship x=−211y. Then, we substitute the values of x and y back into the expression for c and simplify to find the components of c.
Step 7: Calculate the magnitude of c
Finally, we calculate the magnitude of vector c using the formula ∣c∣=cx2+cy2+cz2. We simplify the result.