Find m if the vectors [2,m,5] and [m,-3,7] are perpendicular.
Two vectors are perpendicular if their dot product is zero. The dot product of the vectors $\begin{pmatrix} 2 \\ m \\ 5 \end{pmatrix}$ and $\begin{pmatrix} m \\ -3 \\ 7 \end{pmatrix}$ is $2m - 3m + 35 = -m + 35.$ We set this equal to 0 and solve. $$-m + 35 = 0$$ $$\Rightarrow \quad m = \boxed{35}.$$
Two vectors are perpendicular if their dot product is zero.
The dot product of two vectors [a,b,c] and [d,e,f] is given by:
a*d + b*e + c*f
Let's find the dot product of the two given vectors [2,m,5] and [m,-3,7]:
(2)(m) + (m)(-3) + (5)(7) = 0
Simplifying the equation:
2m - 3m + 35 = 0
Combining like terms:
-m + 35 = 0
To solve for m, we can isolate the variable by subtracting 35 from both sides of the equation:
-m = -35
Now, multiply both sides of the equation by -1 to solve for m:
m = 35
Therefore, m is equal to 35.