How Do You Solve Vector Problems in Triangle Geometry?

[dagg3r]

Vector Gah!

Vector questions. Need help on these three.

In ∆OAB, OA = a and OB = b. P is a point on AB such that BP:PA 1:2.
Q is a point on the extension of OP such that OQ = 1.5OP

a) Express BP and OP in terms of ‘a’ and ‘b’.
these are the answers but how do they get it?
BP = (1/3)(a - b)
OP = (1/3)(a + 2b)

b) Show that AQ = -0.5a + b
i have no idea how to prove this


c) Given that BQ = kOA, find the value of k.
once again this is the answer but i don't know how to approach this can someone show me how to get this value thanks
k = 0.5

 

[Galileo]

Draw the vector that define the triangle. One side is made by the vector a, the other by b and the third by a-b. Then draw the points P and Q. From the drawing you can see that BP is one third the vector (a-b). The others are done similarly.

 

[quicknote]

I would approach these questions by first drawing a diagram of the given information to visualize the triangle and points. Then, I would use the given information and basic geometric principles to solve for the unknown values.

For part a), I would use the fact that BP:PA = 1:2 to set up a ratio equation: BP/PA = 1/2. Since BP = (1/3)(a - b), we can substitute this into the equation to get (1/3)(a - b)/PA = 1/2. We can then solve for PA by cross-multiplying to get PA = (2/3)(a - b). Similarly, using the given information that OQ = 1.5OP, we can set up another ratio equation: OQ/OP = 1.5/1. Using the fact that OQ = 1.5OP, we can substitute this into the equation to get 1.5OP/OP = 1.5/1. This simplifies to 1.5 = 1.5, showing that the given information is consistent. From this, we can solve for OP by setting it equal to any value, such as 1, and solving for it. This gives us OP = 1, and since we know that BP:PA = 1:2, we can substitute this into the equation to get BP = (1/3)(a + 2b).

For part b), we can use the fact that AQ = -0.5a + b and substitute this into the given information that OQ = 1.5OP. This gives us -0.5a + b = 1.5OP = 1.5(1) = 1.5. Solving for AQ, we get AQ = -0.5a + 1.5. To show that this is equal to -0.5a + b, we can substitute in the given information that BQ = kOA. This gives us -0.5a + b = k(a) = ka. Since we know that AQ = -0.5a + 1.5, we can set these two equations equal to each other and solve for k. This gives us -0.5a + 1.5 = ka, which simplifies to k = 0.5.

For part c),