Question Video: Finding an Unknown Term in the Equation of a Plane given the Equation of Its Perpendicular Plane | Nagwa Question Video: Finding an Unknown Term in the Equation of a Plane given the Equation of Its Perpendicular Plane | Nagwa

# Question Video: Finding an Unknown Term in the Equation of a Plane given the Equation of Its Perpendicular Plane Mathematics • Third Year of Secondary School

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Given that the plane 3π₯ β 3π¦ β 3π§ = 1 is perpendicular to the plane ππ₯ β 2π¦ β π§ = 4, find the value of π.

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### Video Transcript

Given that the plane three π₯ minus three π¦ minus three π§ equals one is perpendicular to the plane ππ₯ minus two π¦ minus π§ equals four, find the value of π.

Okay, in this exercise, weβre given equations of two planes and told that these planes are perpendicular to one another. And we can say that if two planes are perpendicular like these are, then it must also be the case that their normal vectors are perpendicular. And this points us to the mathematical condition that is satisfied for two perpendicular planes. The dot product of the normal vectors of these planes must be zero. This suggests that we might solve for the normal vectors of our two given planes and then apply this condition to see if it helps us solve for π. Since both of these equations are given in standard form, we can identify the components of each planeβs normal vector.

For the first plane, its normal vector has an π₯-component of three, a π¦-component of negative three, and a π§-component also of negative three. We can write that like this, calling this normal vector π§ one. For our second plane, its normal vector has an π₯-component of π, a π¦-component of negative two, and a π§-component of negative one. And weβll call this normal vector π§ two. Next, knowing that these two planes are perpendicular, weβll apply this condition. Weβll say that π§ one, this vector here, dotted with π§ two, this vector here, equals zero. If we then carry out this dot product operation, we find that three times π plus six plus three equals zero. And this means that three times the unknown value π is equal to negative nine. Dividing both sides by three, we find that π is equal to negative three. This is the value of π.

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