A fly located at two, four, seven is heading to an open jam jar located at zero, negative two, 10. Given that it will reach the jam jar in four seconds, find the components of its velocity.
We will be able to answer this question using our knowledge of vectors. We know that the position vector of any particle is equal to the vector 𝐫 sub zero plus 𝑡 multiplied by the vector 𝐯, where 𝐫 sub zero is the initial position and 𝐯 is the velocity vector. In this question, we know that the initial position of the fly is equal to two, four, seven. This is our vector 𝐫 sub zero. The end position or location of the jam jar is at zero, negative two, 10. Therefore, this is the value of vector 𝐫. As it takes four seconds for the fly to reach the jam jar, our value of 𝑡 is four. We are trying to calculate vector 𝐯.
Substituting in our values, we have the vector zero, negative two, 10 is equal to the vector two, four, seven plus four multiplied by the vector 𝐯. We can subtract the vector two, four, seven from both sides of this equation. When adding or subtracting two vectors, we simply add or subtract the corresponding components. In this case, we have zero minus two, which is equal to negative two; negative two minus four, which is equal to negative six; and finally 10 minus seven, which is equal to three. The vector negative two, negative six, three is equal to four multiplied by the vector 𝐯.
We can then divide both sides of this equation by four, which is the same as multiplying by one-quarter. Vector 𝐯 is equal to one-quarter multiplied by negative two, negative six, three. When multiplying a vector by a scalar, in this case one-quarter, we need to multiply each of the components by the scalar. One-quarter multiplied by negative two is negative two-quarters or negative one-half. One-quarter multiplied by negative six is negative six-quarters. This simplifies to negative three-halves or negative three over two. Finally, one-quarter multiplied by three is equal to three-quarters.
The velocity vector 𝐯 is equal to negative one-half, negative three over two, and three-quarters. These are the three components of the fly’s velocity.