Chapter 9: Q. 15 (page 627)

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

Short Answer

Expert verified

v ×w= 5i+5j+5k

w ×v= -5i-5j-5k

w ×w=0

v ×v=0

Step by step solution

Step 1. Given information

v= 2i - 3j + k

w = 3i - 2j - k

Step 2. v×w

For two vectors

v=a₁i+b₁j+c₁k

w=a₂i+b₂j+c₂k

cross product v $\times$ 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= 2i - 3j + k

w = 3i - 2j - k

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

Step 3. w ×v

According to the algebraic properties of cross product w $\times$ v= $-$ (v $\times$ w)

so w $\times$ v= $-$ (5i+5j+5k) =-5i-5j-5k

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

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

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

Calculus

Logic and Functions

Statistics

Mechanics Maths

Decision Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

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