Find two vectors parallel to v of the given length. v=\vec{PQ} with P(

Josalynn

Josalynn

Answered question

2021-10-25

Find two vectors parallel to v of the given length.
v=PQ with P(1,7,1) and Q(0,2,5); length=84

Answer & Explanation

krolaniaN

krolaniaN

Skilled2021-10-26Added 86 answers

From the given information, two vectors parallel to v and its length is 84. 
The vector v=PQ with P(1,7,1) and Q(0,2,5). 
As shown, locate the vector v.
v=PQ 
=OQOP 
=(0i+2j+5k)(1i+7j+1k) 
=1i5j+4k 
Hence, the direction of v is <-1,-5,4>. 
Follow these steps to determine the vector's magnitude.
|v|=(1)2+(5)2+42 
=1+25+16 
=42 
Here, the given length is 84. 
Hence, the parallel vectors are 84v|v| and 84v|v| 
Hence, the vector in the direction of v is <8442,42042,33642> and the vector in the opposite direction of v is <8442,42042,33642>
The vector in direction of v is <8442,42042,33642> 
The vector in the opposite direction of v is <8442,42042,33642>.

karton

karton

Expert2023-06-19Added 613 answers

To find two vectors parallel to PQ with length 84, we first calculate the direction vector d of PQ.
Using the coordinates of P(1,7,1) and Q(0,2,5), we can find d as follows:
d=QP=[025][171]=[154]
Next, we normalize d to obtain a unit vector d^:
d^=dd=[154](1)2+(5)2+42=[165623]
Finally, we can obtain two vectors parallel to PQ with length 84 by scaling d^:
v1=84·d^=84·[165623]=[147056]
v2=84·d^=84·[165623]=[147056]
Hence, two vectors parallel to PQ with a length of 84 are v1=[147056] and v2=[147056].
user_27qwe

user_27qwe

Skilled2023-06-19Added 375 answers

Step 1. Calculate the components of v:
v=QP=QxPx,QyPy,QzPz=01,27,51=1,5,4.
Step 2. Normalize v to obtain a unit vector v^:
v=(1)2+(5)2+42=1+25+16=42.
v^=vv=1,5,442=142,542,442.
Step 3. Multiply v^ by the desired length to obtain a vector with the given length:
u1=84·v^=84·142,84·542,84·442=8442,42042,33642.
Step 4. Find a second vector parallel to v by taking the cross product of v and a non-zero vector w:
w=1,0,0 (arbitrarily chosen).
u2=v×w=1,5,4×1,0,0=0,4,5.
Therefore, two vectors parallel to v with a length of 84 are u1=8442,42042,33642 and u2=0,4,5.
alenahelenash

alenahelenash

Expert2023-06-19Added 556 answers

Answer:
v1=84(142(1,5,4))
v2=84(142(1,5,4))
Explanation:
First, we calculate the direction vector d by subtracting the coordinates of point P from those of point Q:
d=QP=(0,2,5)(1,7,1)=(1,5,4)
Next, we find the unit vector u in the direction of d by dividing d by its magnitude:
d=(1)2+(5)2+42=1+25+16=42
u=dd=142(1,5,4)
Finally, we obtain the two vectors parallel to PQ with a length of 84 by scaling the unit vector u:
v1=84u
v2=84u
Thus, the two vectors parallel to PQ with a length of 84 are:
v1=84(142(1,5,4))
v2=84(142(1,5,4))

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