The vectors A⃗ = 1 x^− 2 y^ and B⃗ =− 4 x^+ a y^− 4 z^ are perpendicular to each other. What is the value of a?
a=
To find the value of a, we can use the fact that two vectors are perpendicular if their dot product is zero.
The dot product of two vectors A⃗ = (Ax, Ay, Az) and B⃗ = (Bx, By, Bz) is given by the formula: A⃗ · B⃗ = Ax * Bx + Ay * By+ Az * Bz.
Let's calculate the dot product of A⃗ and B⃗:
A⃗ · B⃗ = (1 * -4) + (-2 * a) + (0 * -4)
= -4 - 2a
Since A⃗ and B⃗ are perpendicular, their dot product must be zero:
-4 - 2a = 0
Now, let's solve the equation for a:
-2a = 4
Dividing both sides of the equation by -2, we get:
a = -2
Therefore, the value of a is -2.