Question Video: Finding the Scalars that Satisfy a Given Operation on Vectors Mathematics • 12th Grade

Given that (3, 𝑦) + π‘₯(3, 5) βˆ’ (0, 7) = 0, find the values of π‘₯ and 𝑦.

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Video Transcript

Given that three, 𝑦 plus π‘₯ multiplied by three, five minus zero, seven is equal to the zero vector, find the values of π‘₯ and 𝑦.

We recall that the zero vector in two dimensions has π‘₯- and 𝑦-coordinate equal to zero. This means that we need to solve the vector equation three, 𝑦 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 𝑦-components, we have 𝑦 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 𝑦 is equal to seven minus five π‘₯. As π‘₯ is equal to negative one, this leaves us with 𝑦 is equal to seven minus five multiplied by negative one. This in turn simplifies to 𝑦 is equal to seven plus five. Adding seven and five gives us 12. The values of π‘₯ and 𝑦 that satisfy the equation are negative one and 12, respectively.

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