Science:Math Exam Resources/Courses/MATH152/April 2010/Question A 29

MATH152 April 2010

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Question A 29
Circle all statements below that are true for any vectors a, b and c in R3. $\mathbf {a} \cdot (\mathbf {a} \times \mathbf {b} )=0$ a, b and c are linearly independent $\mathbf {a} \cdot \mathbf {a} =\|\mathbf {a} \|^{2}$ $\mathbf {a} \cdot \mathbf {b} =0$ $\mathbf {a} \times (\mathbf {b} \times \mathbf {c} )$ is not defined

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If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Recall properties of the dot and cross products such as the dot product of orthogonal vectors is the zero (number) and the cross product of parallel vectors is the zero vector.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. a. $\mathbf {a} \cdot (\mathbf {a} \times \mathbf {b} )=0$ This is true. Recall that the cross product between two vectors $\mathbf {a}$ and $\mathbf {b}$ produces a vector which is orthogonal (perpendicular) to both the original vectors. The dot product between two orthogonal vectors is zero. Therefore since $\mathbf {a} \times \mathbf {b}$ is orthogonal to $\mathbf {a}$ then the dot product should indeed vanish. b. $\mathbf {a} ,\mathbf {b} .\mathbf {c}$ are all linearly independent This is false. We can easily come up with a counter-example of three vectors which are not linearly independent. Consider the vector [1,2,3] and [1,0,0] which are indeed in R3. The vector [3,2,3] is also in R3 but can be written as [1,2,3] + 2 [1,0,0] and therefore is linearly dependent on the other two vectors. c. $\mathbf {a} \cdot \mathbf {a} =||\mathbf {a} ||^{2}$ This is true. Let $\mathbf {a} =[a_{1},a_{2},a_{3}]$ then $\mathbf {a} \cdot \mathbf {a} =a_{1}^{2}+a_{2}^{2}+a_{3}^{2}=||\mathbf {a} ||^{2}$. d. $\mathbf {a} \cdot \mathbf {b} =0$ This is false. We can easily construct two vectors in R3 which do not have a zero dot product. For example consider [1,1,1] and [2,2,2], their dot product is 6. e. $\mathbf {a} \times (\mathbf {b} \times \mathbf {c} )$ is not defined This is false. The cross product is always defined for vectors in R3. The cross product also transforms two vectors in R3 to another vector belonging in R3. Therefore the triple cross product here is just the cross product between two vectors in R3 which is defined. Therefore, only (a) and (c) are true.