Video Transcript
On a lattice, where vector 𝐀𝐂 is
equal to three, three; vector 𝐁𝐂 is equal to 13, negative seven; and two 𝐂 plus
two 𝐀𝐁 is equal to negative four, negative four, find the coordinates of the point
𝐶.
If we begin by considering the
three points 𝐴, 𝐵 and 𝐶 shown on the diagram, we know that vector 𝐀𝐂 is equal
to three, three. This means that we move three units
in the positive 𝑥-direction and three units in the positive 𝑦-direction. Vector 𝐁𝐂 is equal to 13,
negative seven. To travel from point 𝐵 to point
𝐶, we move 13 units in the positive 𝑥-direction and seven units in the negative
𝑦-direction.
We can use this information to find
the vector 𝐀𝐁. One way of traveling from point 𝐴
to point 𝐵 would be via point 𝐶. In order to do this, we would
travel along the vectors 𝐀𝐂 and 𝐂𝐁. We know that vector 𝐀𝐂 is three,
three. Vector 𝐂𝐁 would have the same
magnitude as the vector 𝐁𝐂 but acts in the opposite direction. This means that vector 𝐂𝐁 is
equal to negative 13, seven. Vector 𝐀𝐁 is therefore equal to
three, three plus negative 13, seven.
We know that we can add two vectors
by adding their corresponding components. Three plus negative 13 is equal to
negative 10, and three plus seven is 10. Therefore, vector 𝐀𝐁 is equal to
negative 10, 10. If we let the point 𝐶 have
coordinates 𝑥, 𝑦, then the position vector of point 𝐶, also written 𝐎𝐂, is
equal to the vector 𝑥, 𝑦. Substituting this together with the
vector 𝐀𝐁 into the equation given, we have two multiplied by 𝑥, 𝑦 plus two
multiplied by negative 10, 10 is equal to negative four, negative four.
We recall that we can multiply a
vector by a scalar by multiplying each of the components by that scalar. Our equation simplifies to two 𝑥,
two 𝑦 plus negative 20, 20 is equal to negative four, negative four. We can add the two vectors on the
left-hand side such that two 𝑥 minus 20, two 𝑦 plus 20 is equal to negative four,
negative four. And finally, to calculate the
values of 𝑥 and 𝑦, we can equate the corresponding components. We have two equations: two 𝑥 minus
20 is equal to negative four, and two 𝑦 plus 20 is equal to negative four. Solving the first equation gives us
𝑥 is equal to eight. And solving the second equation, we
have 𝑦 is equal to negative 12. We can therefore conclude that the
coordinates of point 𝐶 are eight, negative 12.
