The dot product of u with itself is 12. What is the magnitude of u?
A 24
B 12
C 2sqrt6
D 2sqrt3
How would you solve this, what is the formlua?
a ⋅ b = a * b * cos(Θ) ... Θ is the angle between a and b
u ⋅ u = u * u * cos(0) ... the angle between u and itself is zero
u ⋅ u = u^2 = 12 ... solve for u
Well, to find the magnitude of vector u, you can use the formula ||u|| = √(u · u), where · represents the dot product. In this case, since the dot product of u with itself is given as 12, we can plug it into the formula as follows:
||u|| = √(12)
Now let's simplify this equation and unveil the magnificent answer. Drumroll, please! 🥁
To find the magnitude of vector u, we use the formula:
|u| = √(u · u)
Where u · u represents the dot product of vector u with itself.
Given that the dot product of u with itself is 12:
|u| = √(12)
Simplifying further:
|u| = √(4 × 3)
|u| = √4 × √3
|u| = 2√3
Therefore, the magnitude of vector u is option D) 2√3.
To solve this problem, we need to find the magnitude of vector u. The dot product of a vector with itself is equal to the square of its magnitude.
The formula for the dot product of two vectors u = (u1, u2, u3) and v = (v1, v2, v3) is given by:
u · v = u1v1 + u2v2 + u3v3
In this case, the dot product of u with itself is 12, so we have:
u · u = u1u1 + u2u2 + u3u3 = 12
Since we want to find the magnitude of u, which is denoted as |u|, we can rewrite the equation as:
|u|^2 = 12
To find |u|, we take the square root of both sides:
|u| = sqrt(12)
Simplifying the square root expression, we get:
|u| = sqrt(4 * 3)
Since 4 is a perfect square, we can simplify further:
|u| = 2sqrt(3)
Therefore, the magnitude of vector u is 2sqrt(3). The answer is option D.