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Chapter 10: Q13RE (page 611)

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

For any vectors \({\rm{u}}\)and \({\rm{v}}\)in\({{\rm{V}}_{\rm{3}}}\), \(\left( {{\rm{u}} \times {\rm{v}}} \right) \cdot {\rm{u = 0}}\)

Short Answer

Expert verified

The stated statement is true.

Step by step solution

01

Check to see if the statement is true or not.

True; because the cross product is a vector that is perpendicular to two vectors, it must also be perpendicular to one of them (i.e dot product is zero). The cross product, in reality, is perpendicular to the plane, and any vectors located on the plane that joins the two vectors are cross multiplied.

02

Result.

True; because the cross product is a vector that is perpendicular to two vectors, it must also be perpendicular to one of them (i.e dot product is zero).

Therefore, the stated statement is true.

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Most popular questions from this chapter

Find the cross product between \({\rm{a}}\) and \({\rm{b}}\) and \({\rm{b}}\) and \({\rm{a}}\).

Let \({\bf{v}} = 5{\bf{j}}\) and let \({\bf{u}}\) be a vector with length \(3\) that starts at the origin and rotates in the \(xy\)-plane.

(a) Find the maximum values of the length of the vector \({\bf{u}} \times {\bf{v}}\).

(b) Find the minimum values of the length of the vector \({\bf{u}} \times {\bf{v}}\).

(c) In what direction does \({\bf{u}} \times {\bf{v}}\) point?

To determine whether the given expression is meaningful or meaningless.

(a) \(({\rm{a}} \cdot {\rm{b}}) \cdot {\rm{c}}\)

(b) \((a \cdot b)c\)

(c) \(|{\rm{a}}|({\rm{b}} \cdot {\rm{c}})\)

(d) \(a \cdot (b + c)\)

(e) \(a \cdot b + c\)

(f) \(|a| \cdot (b + c)\)

To show

(a) \({\rm{i}} \cdot {\rm{j}} = 0,{\rm{j}} \cdot {\rm{k}} = 0\) and \({\rm{k}} \cdot {\rm{i}} = 0\).

(b) \({\rm{i}} \cdot {\rm{i}} = 1,{\rm{j}} \cdot {\rm{j}} = 1\) and \({\rm{k}} \cdot {\rm{k}} = 1\)

(a) Find the symmetric equations for the line that passes through the point \((1, - 5,6)\) and parallel to the vector \(\langle - 1,2, - 3\rangle \).

(b) Find the point at which the line (that passes through the point \((1, - 5,6)\) and parallel to the vector \(\langle - 1,2, - 3\rangle )\) intersects the \(xy\)-plane, the point at which the line (that passes through the point \((1, - 5,6)\) and parallel to the vector \(\langle - 1,2, - 3\rangle )\) intersects the \(yz\)-plane and the point at which the line (that passes through the point \((1, - 5,6)\) and parallel to the vector \(\langle - 1,2, - 3\rangle )\) intersects the \(xz\)-plane.
See all solutions

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