Chapter 9: Q. 44 (page 630)

In Problems 40–45, use the vectors $v = 3 i + j - 2 k$ and $w = - 3 i + 2 j - k$ to find each expression.
$v \cdot (v \times w)$

Short Answer

Expert verified

The expression $v \cdot (v \times w) = 0 .$

Step by step solution

Step 1. Given Information

The given vectors are $v = 3 i + j - 2 k a n d w = - 3 i + 2 j - k .$

Step 2. Finding the given expression

To find the expression, substitute the values of vectors in the given expression.

Thus, $v \cdot (v \times w) = (3 i + j - 2 k) \cdot ((3 i + j - 2 k) \times (- 3 i + 2 j - k)) .$

Now, let's first find the cross product, by using the definition of it.

If $v = a_{1} i + b_{1} j + c_{1} k$ and $w = a_{2} i + b_{2} j + c_{2} k$ are two vectors in space, the

cross product $v \times w$ is defined as the vector

$v \times w = (b_{1} c_{2} - b_{2} c_{1}) i - (a_{1} c_{2} - a_{2} c_{1}) j + (a_{1} b_{2} - a_{2} b_{1}) k .$

Step 3. Solve

By proceeding with the calculation further using the formula of the cross product.

So,

$v \times w = (1 \cdot (- 1) - 2 \cdot (- 2)) i - (3 \cdot (- 1) - (- 3) \cdot (- 2)) j + (3 \cdot 2 - (- 3) \cdot 1) k v \times w = 3 i + 9 j + 9 k$

Let's find the dot product, the formula of the dot product is $v \cdot w = a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} .$

So,

$v \cdot (v \times w) = (3 i + j - 2 k) \cdot (3 i + 9 j + 9 k) v \cdot (v \times w) = (3 \cdot 3) + (1 \cdot 9) + (- 2 \cdot 9) v \cdot (v \times w) = 9 + 9 - 18 v \cdot (v \times w) = 0$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

In Problems 40–45, use the vectors $v = 3 i + j - 2 k$ and $w = - 3 i + 2 j - k$ to find each expression.
$v \cdot (v \times w)$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding the given expression

Step 3. Solve

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Mechanics Maths

Theoretical and Mathematical Physics

Statistics

Pure Maths

Calculus

Study anywhere. Anytime. Across all devices.

In Problems 40–45, use the vectors v=3i+j-2kand w=-3i+2j-kto find each expression. v·v×w

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding the given expression

Step 3. Solve

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Mechanics Maths

Theoretical and Mathematical Physics

Statistics

Pure Maths

Calculus

Study anywhere. Anytime. Across all devices.

In Problems 40–45, use the vectors $v = 3 i + j - 2 k$ and $w = - 3 i + 2 j - k$ to find each expression.
$v \cdot (v \times w)$