Chapter 9: Q. 16 (page 627)

find (a) v × w, (b) w × v, (c) w × w, and
(d) v × v.
v = -i + 3j + 2k
w = 3i - 2j - k

Short Answer

Expert verified

v ×w= i +5j -7k

w ×v= -i -5j +7k

w ×w=0

v ×v=0

Step by step solution

Step 1. Given information

v = -i + 3j + 2k

w = 3i - 2j - k

Step 2. v× w

For two vectors

v=a₁i+b₁j+c₁k

w=a₂i+b₂j+c₂k

cross product v× w is written as

$v \times w = |\begin{matrix} i & j & k \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{matrix}| = i |\begin{matrix} b_{1} & c_{1} \\ b_{2} & c_{2} \end{matrix}| - j |\begin{matrix} a_{1} & c_{1} \\ a_{2} & c_{2} \end{matrix}| + k |\begin{matrix} a_{1} & b_{1} \\ a_{2} & b_{2} \end{matrix}|$

v = -i + 3j + 2k

w = 3i - 2j - k

$v \times w = |\begin{matrix} i & j & k \\ - 1 & 3 & 2 \\ 3 & - 2 & - 1 \end{matrix}| = i |\begin{matrix} 3 & 2 \\ - 2 & - 1 \end{matrix}| - j |\begin{matrix} - 1 & 2 \\ 3 & - 1 \end{matrix}| + k |\begin{matrix} - 1 & 3 \\ 3 & - 2 \end{matrix}| = i + 5 j - 7 k$

Step 3. w × v

According to the algebraic properties of cross product

w × v = -(v × w)

so w × v= $-$ (i+5j-7k)= -i -5j +7k

Step 4. w× w and v× v

According to the algebraic properties of cross product the cross product of any vector with itself is the zero vector

(a ×a=0).

Hence w ×w=0 and v ×v=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.

find (a) v × w, (b) w × v, (c) w × w, and
(d) v × v.
v = -i + 3j + 2k
w = 3i - 2j - k

Short Answer

Step by step solution

Step 1. Given information

Step 2. v× w

Step 3. w × v

Step 4. w× w and v× v

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Statistics

Discrete Mathematics

Theoretical and Mathematical Physics

Applied Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.

find (a) v × w, (b) w × v, (c) w × w, and(d) v × v.v = -i + 3j + 2kw = 3i - 2j - k

Short Answer

Step by step solution

Step 1. Given information

Step 2. v× w

Step 3. w × v

Step 4. w× w and v× v

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Statistics

Discrete Mathematics

Theoretical and Mathematical Physics

Applied Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.

find (a) v × w, (b) w × v, (c) w × w, and
(d) v × v.
v = -i + 3j + 2k
w = 3i - 2j - k