 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2021-resit/9/i/solution

Use

w = - u - v

v \times w = - v \times u - v \times v = u \times v, w \times u = - u \times u - v \times u = u \times v .

Consequently,

u \times v = v \times w = w \times u, span {u, v, w} = span {u, v} .

u, v

u \times v

. If they are dependent, all three nonzero vectors lie on one line through the origin, and their linear span has dimension one. For example,

u = v = (1, 0, 0)

and

w = (- 2, 0, 0)

obey the hypothesis but span only a line. Nonzero vectors alone do not force a two-dimensional linear span.

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