The notation "||v||" typically refers to the Euclidean norm (also known as the magnitude or length) of a vector v in a Euclidean space. The Euclidean norm of a vector v in n-dimensional space is defined as the square root of the sum of the squares of its components:
||v|| = sqrt(v1^2 + v2^2 + ... + vn^2)
Given that ||v|| = 8, we can find the Euclidean norm of -7v as follows:
||-7v|| = 7 * ||-v|| = 7 * sqrt((-v1)^2 + (-v2)^2 + ... + (-vn)^2)
Since -v has the same magnitude as v but points in the opposite direction, we can replace each vi in the above expression with -vi:
||-7v|| = 7 * sqrt((-v1)^2 + (-v2)^2 + ... + (-vn)^2) = 7 * sqrt(v1^2 + v2^2 + ... + vn^2)
But we know that ||v|| = sqrt(v1^2 + v2^2 + ... + vn^2) = 8, so we can substitute:
||-7v|| = 7 * ||v|| = 7 * 8 = 56
Therefore, ||-7v|| is 56.