Determine the magnitude of the vector v=(3,4)?(K:1)

Question

BlackTom AI · Accepted Answer

We’re asked to find the magnitude (length) of the vector v = (3, 4). To understand what this means, recall that the magnitude of a 2D vector (x, y) is computed via the Pythagorean theorem as sqrt(x^2 + y^2).
Option a. 7: If the magnitude were 7, then x^2 + y^2 would have to equal 49. For (3, 4), x^2 + y^2 = 9 + 16 = 25, which is not 49, so this is incorrect.
Option b. 4: If the magnitude were 4, then x^2 + y^2 would have to equal 16. But 9 + 16 = 25, not 16, so this cannot be the magnitude.
Option c. 5: If the magnitude were 5, then x^2 + y^2 would have to equal 25. Indeed, 3^2 + 4^2 = 9 + 16 = 25, and sqrt(25) = 5, so this matches perfectly.
Option d. 3: If the magnitude were 3, then x^2 + y^2 would have to equal 9. Since 9 + 16 = 25, this would not be consistent with a magnitude of 3, making this option incorrect.
In summary, the only choice that satisfies the magnitude formula for v = (3, 4) is 5, because the squared components sum to 25, whose square root is 5.

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