$ \mathrm{A}^{\rightarrow}=3 \hat{\imath}+2 \hat{\jmath}-5 \mathrm{k}^{\wedge}\( and \)\mathrm{B}^{\rightarrow}=\hat{\imath}+\hat{\jmath}+2 \mathrm{k}^{\wedge}\( Find \)\left|\mathrm{A}^{\rightarrow}+2 \mathrm{~B}^{\rightarrow}\right|$ (A) \(\sqrt{40}\) (B) \(\sqrt{42}\) (C) \(\sqrt{39}\) (D) 2

Short Answer

Expert verified
(B) \(\sqrt{42}\)

Step by step solution

01

Multiply vector B by 2

To find 2B, we multiply each component of vector B by 2: 2B = 2(1, 1, 2) = (2, 2, 4)
02

Add vector A and the 2B

Now, add the components of vector A and vector 2B: A + 2B = (3, 2, -5) + (2, 2, 4) = (3 + 2, 2 + 2, -5 + 4) = (5, 4, -1)
03

Find the magnitude of the resulting vector

Find the magnitude of the resultant vector A + 2B = (5, 4, -1) using the formula for magnitude: \(|A+2B| = \sqrt{(5^2) + (4^2) + (-1^2)}\)
04

Calculate the magnitude

Now, calculate the magnitude of the resultant vector: \(|A+2B| = \sqrt{(25) + (16) + (1)} = \sqrt{42}\) After solving for the magnitude, it matches option B, so the correct answer is: (B) \(\sqrt{42}\)

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