MHB #23 intersecting vector equations of 2 lines

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To find the intersection point of two lines represented by vector equations, converting them to the slope-intercept form (y = mx + b) and solving simultaneously is one effective method. The parameters λ and t are used to traverse the lines, but they do not directly represent the intersection point. By substituting λ = -1 or t = 1 into the respective equations, the coordinates of the intersection can be determined. The calculations show that the intersection point P is (2, 3). This approach highlights the importance of manipulating the equations to find the solution accurately.
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the calcs are mine, the only way I could see to solve this was to convert these to the $y=mx+b$ line eq and solve simultaneously to the intersection of $(x,y)$

not sure how you take them as they are given and find point P the intersection just as vectors.

also, I tried to input a vector equation of a line in W|A but my syntax didn't produce the correct line. or is there a an input for that.
 
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Try this, yielding $P=\langle 2,3 \rangle$.
 
nice to know that

so $(\lambda,\tau)$ is the intersection?
 
No, it is not. $\lambda$ and $t$ are two parameters that, when you change them, help you to traverse the two lines given by the two vector equations you have. To find the point, plug in $\lambda=-1$ into the first equation, or $t=1$ into the second, and the coordinates you get there will be what you need.
 
Another approach would be to write the system:

$$5+3\lambda=-2+4t$$

$$1-2\lambda=2+t$$

simplify:

$$4t-3\lambda=7$$

$$t+2\lambda=-1$$

Subtracting 4 times the latter equation from the former, we find:

$$-11\lambda=11\implies\lambda=-1$$

Hence:

$$P=(5+3(-1),1-2(-1))=(2,3)$$
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
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