Video Transcript
In the triangle π΄π΅πΆ, π· lies on
the line segment π΅πΆ, where the ratio of π΅π· to π·πΆ is two to three. Given that three multiplied by
vector ππ plus two multiplied by vector ππ is equal to π multiplied by vector
ππ, find the value of π.
We will begin by considering the
triangle π΄π΅πΆ. We know that point π· lies on π΅πΆ
and the ratio of π΅π· to π·πΆ is two to three. This means that the vector ππ is
two-fifths of the vector ππ. Letβs now consider the equation
weβre given. Three multiplied by vector ππ
plus two multiplied by vector ππ is equal to π multiplied by vector ππ. We know that vector ππ is equal
to ππ plus ππ. This means that the left-hand side
of our equation can be rewritten as three ππ plus two multiplied by ππ plus
ππ. We can distribute the parentheses
to get two ππ plus two ππ. Collecting like terms, we have five
ππ plus two ππ.
Letβs now consider the right-hand
side of the equation. Vector ππ is equal to ππ plus
ππ. Therefore, the right-hand side is
equal to π multiplied by ππ plus ππ. We know that ππ is equal to
two-fifths of ππ. We can then distribute the
parentheses. This gives us π multiplied by
vector ππ plus two-fifths π multiplied by vector ππ. We can now equate coefficients. Five must be equal to π. Two must be equal to two-fifths of
π. Dividing both sides of this
equation by two-fifths also gives us π is equal to five. The value for π such that three
multiplied by vector ππ plus two multiplied by vector ππ is equal to π
multiplied by vector ππ is five.