The plane $r.(a,3,5)=10$ is inclined at an angle of $45^\circ$ to the plane $r.(-5,1,4)$
Find the value(s) of $a$ up to $2$ decimal places.
I attempted this problem by forming an equation where I let the dot product of the two normal vectors divided by the magnitude of each of the normal vectors multiplied together equal to $\cos 45^\circ$ I tried to solve it on my graphics calculator, but it didn't seem to be able to find a solution.
How would one go about solving this problem??
You are on the right track. Squaring both sides of $$(a,3,5)\cdot(-5,1,4)=\pm\|(a,3,5)\|\|(-5,1,4)\|\cos(45^\circ),$$ you get a quadratic equation in $a$.