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Question
For `vec v_1 = 2 hat i - 3 hat j` and `vec v_2 = -6hat i + 5 hat j`, determine the magnitude and direction of `vec v_1 + vec v_2`.
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Solution
`vec v_1 + vec v_2 = (2 hat i - 3 hat j) + (-6 hat i + 5 hat j)`
= `(2 hat i - 6 hat i) + (-3 hat j + 5 hat j)`
= `-4 hat i + 2 hat j`
∴ `|vec v_1 + vec v_2| = sqrt((-4)^2 + 2^2)`
= `sqrt 20`
= `sqrt(4 xx 5)`
= `2 sqrt 5`
Comparing `vec v_1 + vec v_2` with `vec R = R_x hat i + R_y hat j`
⇒ Rx = −4 and Ry = 2
Taking θ to be angle made by `vec R` with X – axis,
∴ θ = `tan^-1(R_y/R_x)`
= `tan^-1 (2/-4)`
= `tan^-1 (- 1/2)` with X-axis
∴ Magnitude and direction of `vec v_1 + vec v_2` is respectively `2 sqrt 5` and `tan^-1 (- 1/2)` with the X-axis.
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