Video Transcript
Given that three, π¦ plus π₯ multiplied by three, five minus zero, seven is equal to the zero vector, find the values of π₯ and π¦.
We recall that the zero vector in two dimensions has π₯- and π¦-coordinate equal to zero. This means that we need to solve the vector equation three, π¦ plus π₯ multiplied by three, five minus zero, seven is equal to zero, zero. We begin by multiplying the vector three, five by the scalar π₯. π₯ multiplied by three is equal to three π₯, and π₯ multiplied by five is equal to five π₯.
Our next step is to recall that when adding or subtracting vectors, we simply add or subtract the corresponding components. Equating the π₯-components, we have three plus three π₯ minus zero is equal to zero. Subtracting three from both sides of this equation gives us three π₯ is equal to negative three. We can then divide both sides of this equation by three, giving us a value of π₯ equal to negative one.
Equating the π¦-components, we have π¦ plus five π₯ minus seven is equal to zero. We can add seven and subtract five π₯ from both sides of this equation so that π¦ is equal to seven minus five π₯. As π₯ is equal to negative one, this leaves us with π¦ is equal to seven minus five multiplied by negative one. This in turn simplifies to π¦ is equal to seven plus five. Adding seven and five gives us 12. The values of π₯ and π¦ that satisfy the equation are negative one and 12, respectively.
