Physics: Vectors & their scalar product

[sakau2007]

Homework Statement

Given the vectors:
P = 8i +5j-P_zk m
and
Q = 3i -4j-2k m
Determine the value of P_z so that the scalar product of the two vectors will be 60m²

Homework Equations

Sure seems like we will need to use the following equation:
P * Q = |P| * |Q| * cos ∅

But I don't recall being able to be taught the angle between two vectors so cos would still be unknown?

The Attempt at a Solution

Also, I assume the scalar product being 60 means that PQ = 60, or that:
Q = √3²+-4²+2² = √29
So, P would be 60/√29

Then this equation should be solved such as
60/√29 = √8²+5²+(-P_z)²

11.14172029 = √8²+5²+(-P_z)²

124.13793 = 89 + (-P_z)²

35.13793 = (-P_z)²

This would give an answer of about +/- 5.92, but in my solution manual I see that the solution is -28. Where have I gone wrong?

 

[consciousness]

Your job would be simplified by the "Distributive property of dot product".

 

[Chestermiller]

Here's a hint:

The scalar product of i with i is equal to ??

The scalar product of i with j is equal to ??

The scalar product of i with k is equal to ??

 

[sakau2007]

It was me not understanding what scalar product was in solving this.

When the scalar product of PQ is said to be 60, that means that:
(8*3) + (5*-4)+(2*-P_z) = 60
Solving that simple equation yields the correct answer of -28.