Final answer:
The magnitude of the vector C, which satisfies the vector equation 2A – 6B + 3C = 2j with given vectors A and B, is calculated using the components of C and the formula for vector magnitude, resulting in |C| = √8.
Step-by-step explanation:
To find the magnitude of the vector C that satisfies the vector equation 2A – 6B + 3C = 2j, we first need to express vectors A and B in their component form. Given that vector A = –i – 2k and vector B = –j + k/2, we can substitute these into the equation and solve for vector C's components.
After substituting A and B, we have 2(-i – 2k) – 6(-j + k/2) + 3C = 2j. Simplifying this, we get – 2i – 4k + 6j – 3k + 3C = 2j. Combining like terms and isolating C gives us 3C = 2i + 8k – 6j + 2j, resulting in 3C = 2i + 2j + 8k. To find C, we divide each component by 3, giving C = (2/3)i + (2/3)j + (8/3)k.
Finally, to find the magnitude of C, we use the formula √(i-component)^2 + (j-component)^2 + (k-component)^2, leading to |C| = √((2/3)^2 + (2/3)^2 + (8/3)^2) = √(4/9 + 4/9 + 64/9) = √(72/9) = √8.