Find the vector 𝐀𝐆 using the
One way of answering this question
would be to recall that we can find the vector 𝐀𝐁 by subtracting vector 𝐀 from
vector 𝐁. This means that in our question, we
need to subtract vector 𝐀 from vector 𝐆. Vector 𝐀 is the displacement of
point 𝐴 from the origin. This has an 𝑥-component of one, a
𝑦-component of one, and a 𝑧-component of zero. This means that vector 𝐀 is equal
to one, one, zero. Vector 𝐆 has an 𝑥-component of
four, a 𝑦-component of four, and a 𝑧-component of three. This means that vector 𝐆 is equal
to four, four, three.
To calculate vector 𝐀𝐆, we need
to subtract one, one, zero from four, four, three. When subtracting vectors, we
subtract each component separately. Four minus one is equal to
three. When subtracting the 𝑦-components,
we also get three. The same is true of the
𝑧-components as three minus zero is equal to three. Vector 𝐀𝐆 is, therefore, equal to
three, three, three.
An alternative method here would be
to recognize that we have a cube of side length three. Vertices 𝐴 and 𝐺 are opposite
corners of the cube. This means that we need to move
three units in the 𝑥-, 𝑦-, and 𝑧-direction to get from point 𝐴 to point 𝐺. This confirms that vector 𝐀𝐆 is
equal to three, three, three.