Show full answers Determine whether $\vec{u} \perp \vec{v}$. (Notation $\perp$ means perpendicular; i.e., $\vec{u} \perp \vec{v}$ means $\vec{u}$ and $\vec{v}$ are orthogonal.)(a) $\vec{u} = 7\vec{i} + 3\vec{j} + 5\vec{k}$, $\vec{v} = -8\vec{i} + 4\vec{j} + 2\vec{k}$.(b) $\vec{u} = 6\vec{i} + \vec{j} + 3\vec{k}$, $\vec{v} = 4\vec{i} - 6\vec{k}$.(c) $\vec{u} = \langle 1, 1, 1 \rangle$, $\vec{v} = \langle -1, 0, 0 \rangle$.

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Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Determine whether $\vec{u} \perp \vec{v}$. (Notation $\perp$ means perpendicular; i.e., $\vec{u} \perp \vec{v}$ means $\vec{u}$ and $\vec{v}$ are orthogonal.)

(a) $\vec{u} = 7\vec{i} + 3\vec{j} + 5\vec{k}$, $\vec{v} = -8\vec{i} + 4\vec{j} + 2\vec{k}$.

(b) $\vec{u} = 6\vec{i} + \vec{j} + 3\vec{k}$, $\vec{v} = 4\vec{i} - 6\vec{k}$.

(c) $\vec{u} = \langle 1, 1, 1 \rangle$, $\vec{v} = \langle -1, 0, 0 \rangle$.

Expert Solution

Step 1: Part(a)

If the vector two vectors are orthogonal then their dot product is zero.

The given vectors are:

$u equals open parentheses table row 7 3 5 end table close parentheses comma thin space thin space v equals open parentheses table row cell negative 8 end cell 4 2 end table close parentheses$

Now find the dot product.

$u times v equals open parentheses table row 7 3 5 end table close parentheses times open parentheses table row cell negative 8 end cell 4 2 end table close parentheses u times v equals 7 open parentheses negative 8 close parentheses plus 3 times thin space 4 plus 5 times thin space 2 u times v equals negative 56 plus 12 plus 10 u times v equals negative 34 u times v not equal to 0$

Here the dot product is not zero.

Therefore the given vectors are orthogonal.