Let u = [2,−3, 1] and v = [0, 1, 4]. Calculate the vector u − v. Place u and v tail-totail to form two sides of a triangle. With regard to this triangle, what does u−v represent?
Calculate the number u•u and discuss its relevance to the diagram you have drawn. Do the same for the number
(u − v) • (u − v).
To calculate the vector u - v, we need to subtract the corresponding components of u and v.
u - v = [2, -3, 1] - [0, 1, 4] = [2 - 0, -3 - 1, 1 - 4] = [2, -4, -3].
Now, let's place u and v tail-to-tail to form two sides of a triangle.
To do this, we start with the tail of vector v at the head of vector u, and draw vector u-v from the tail of u to the head of v. The resulting diagram should show the three sides of a triangle.
Now let's discuss what u-v represents in this triangle. The vector u-v represents the vector connecting the opposite vertex of the triangle formed by u, v, and u-v. In other words, it represents the side of the triangle opposite to the angle formed by vectors u and v.
Next, let's calculate the dot product of vector u with itself, u • u, and discuss its relevance to the diagram.
The dot product of a vector with itself is the square of its magnitude (length). Therefore, u • u = ||u||^2 = (2^2 + (-3)^2 + 1^2) = (4 + 9 + 1) = 14.
In the diagram, the squared magnitude of u, ||u||^2 = 14, represents the sum of the squares of the lengths of the sides adjacent to the angle formed by u and v. This is a useful measurement in various geometrical calculations and can provide information about the size and shape of the triangle.
Finally, let's calculate (u - v) • (u - v) and discuss its relevance to the diagram.
To calculate (u - v) • (u - v), we need to take the dot product of vector u - v with itself.
(u - v) • (u - v) = ||u - v||^2 = (2^2 + (-4)^2 + (-3)^2) = (4 + 16 + 9) = 29.
In the diagram, ||u - v||^2 = 29 represents the squared magnitude of the vector u - v. This is useful in determining the length of the side opposite to the angle formed by u and v in the triangle.
So, (u - v) • (u - v) provides information about the length and size of the triangle formed by u, v, and u - v.