Here are a couple of true statements. Make sure that you understand why all these statements are true. Once you do click "I get it :)". Cross your heart and ... :) The best way to make sense of these statements is to translate them into geometry. Okay, so let [math: u]{\bf u} and [math: v] {\bf v} be vectors in R3. The cross product only makes sense for vectors in R3.[math: u×v+v×u=0] {\bf u} imes {\bf v} +{\bf v} imes {\bf u} = {\bf 0}[math: (3u)×v=3(u×v)] (3 {\bf u}) imes {\bf v}= 3({\bf u} imes {\bf v} )If [math: u] {\bf u} is a multiple of [math: v] {\bf v} , then [math: u×v=0]{\bf u} imes {\bf v} = {\bf 0}[math: u×u=0] {\bf u} imes {\bf u} = {\bf 0}

Question

Here are a couple of true statements. Make sure that you understand why all these statements are true. Once you do click "I get it :)". Cross your heart and ... :) The best way to make sense of these statements is to translate them into geometry. Okay, so let [math: u]{\bf u} and [math: v] {\bf v} be vectors in R3. The cross product only makes sense for vectors in R3.[math: u×v+v×u=0] {\bf u} 	imes {\bf v} +{\bf v} 	imes {\bf u} = {\bf 0}[math: (3u)×v=3(u×v)] (3 {\bf u}) 	imes {\bf v}= 3({\bf u} 	imes {\bf v} )If [math: u] {\bf u} is a multiple of [math: v] {\bf v} , then [math: u×v=0]{\bf u} 	imes {\bf v} = {\bf 0}[math: u×u=0] {\bf u} 	imes {\bf u} = {\bf 0}

BlackTom AI · Accepted Answer

The task presents a collection of statements about vectors in R3 and the cross product, and asks us to determine understanding.

First, assess the mathematical content of the statements themselves. The cross product is anti-commutative, meaning u × v = - (v × u); therefore u × v + v × u = 0 is a correct reflection of that property.

Next, scalar multiplication behaves with the cross product: (3u) × v = 3 (u × v). This follows from the bilinear property of the cross product, where scaling one argument scales the result by the same factor.

Then, if u is a multiple of v, their cross product vanishes: u × v = 0 when u and v are parallel. This is because the vectors lie on the same or opposite line, so the area of the parallelogram spanned by them is zero.

Also, the cross product of a vector with itself is zero: u × u = 0. This is a direct consequence of the anti-symmetric nature of the cross product, since swapping the two identical vectors would change the sign, implying the vector must be zero.

The initial statement about the cross product only making sense in R3 is a traditional caveat; while the cross product in literature is defined for three-dimensional space, similar constructions exist in higher dimensions (with different definitions), but in the standard context used here, R3 is the setting.

Now, regarding the question structure: the provided answer field lists 'b. I get it :)' but the answer_options array is empty, so there are no explicit choices to evaluate. This mismatch means we cannot map any option letters (a, b, c, …) to a stated truth value within the given multiple-choice format.

Given the content, if this were a standard single-select item with explicit options, we would expect all the statements identified above to be true, making any option asserting that all listed statements are true a correct choice. The absence of answer options prevents a formal selection from being judged in this prompt.

In summary, the mathematical claims embedded in the statements are individually true: (1) u × v + v × u = 0, (2) (3u) × v = 3(u × v), (3) if u is a multiple of v then u × v = 0, and (4) u × u = 0. The missing answer_options block means we cannot assign a specific option label to those conclusions within a multiple-choice framework here. If you intended to mark understanding with the option 'I get it :)', that suggests you acknowledge comprehension, but the configuration of options does not provide a concrete target to select.

MTH1030 -1035 - S1 2025 MTH1030/5 MTH1030/5 Week 1 lesson quiz: Vectors, lines and planes

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