Chapter 10: Q. 13 (page 824)

Give an example of three nonzero vectors u, v and w in $ℝ^{3}$ such that $u \times v = u \times w$ but $v \neq w$ . What geometric relationship must the three vectors have for this to happen?

Short Answer

Expert verified

Let $u = (1, 0, 0), v = (2, 1, 1) and w = (4, 1, 1)$ .

If $u \times v = u \times w$ , then u is parallel to $v - w$ .

Step by step solution

Step 1. Given Information

Give an example of three nonzero vectors u, v and w in $ℝ^{3}$ such that role="math" localid="1649322061069" $u \times v = u \times w$ but $v \neq w$ . What geometric relationship must the three vectors have for this to happen?

Step 2. Let u=(1,0,0), v=(2,1,1) and w=(4,1,1)

Now finding the value of $u \times v$ .

$u \times v = \det [\begin{matrix} i & j & k \\ 1 & 0 & 0 \\ 2 & 1 & 1 \end{matrix}] u \times v = ((0) (1) - (1) (0)) i + ((1) (1) - (2) (0)) j + ((1) (1) - (2) (0)) k u \times v = (0 + 0) i + (1 - 0) j + (1 - 0) k u \times v = 0 i + 1 j + 1 k$

Step 3. Now finding the value of

$u \times w = \det [\begin{matrix} i & j & k \\ 1 & 0 & 0 \\ 4 & 1 & 1 \end{matrix}] u \times w = ((0) (1) - (1) (0)) i + ((1) (1) - (4) (0)) j + ((1) (1) - (4) (0)) k u \times w = (0 + 0) i + (1 - 0) j + (1 - 0) k u \times w = 0 i + 1 j + 1 k$

Hence, $u \times v = u \times w = 0 i + 1 j + 1 k$ but $v \neq w$ .

Step 4. Now finding the relation of three vectors.

If $u \times v = u \times w$ , then u is parallel to $v - w$ .

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Give an example of three nonzero vectors u, v and w in $ℝ^{3}$ such that $u \times v = u \times w$ but $v \neq w$ . What geometric relationship must the three vectors have for this to happen?

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Let u=(1,0,0), v=(2,1,1) and w=(4,1,1)

Step 3. Now finding the value of

Step 4. Now finding the relation of three vectors.

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Give an example of three nonzero vectors u, v and w in ℝ3 such that u×v=u×w but v≠w. What geometric relationship must the three vectors have for this to happen?

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Let u=(1,0,0), v=(2,1,1) and w=(4,1,1)

Step 3. Now finding the value of

Step 4. Now finding the relation of three vectors.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Calculus

Statistics

Theoretical and Mathematical Physics

Discrete Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Give an example of three nonzero vectors u, v and w in $ℝ^{3}$ such that $u \times v = u \times w$ but $v \neq w$ . What geometric relationship must the three vectors have for this to happen?