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

Question

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

krolaniaN · Accepted Answer

From the given information, two vectors parallel to v and its length is 84.&amp;nbsp;The vector&amp;nbsp;v=PQ→&amp;nbsp;with P(1,7,1) and Q(0,2,5).&amp;nbsp;As shown, locate the vector v.v=PQ→&amp;nbsp;=OQ→−OP→&amp;nbsp;=(0i+2j+5k)−(1i+7j+1k)&amp;nbsp;=−1i−5j+4k&amp;nbsp;Hence, the direction of v is &amp;lt;-1,-5,4&amp;gt;.&amp;nbsp;Follow these steps to determine the vector&#039;s magnitude.|v|=(−1)2+(−5)2+42&amp;nbsp;=1+25+16&amp;nbsp;=42&amp;nbsp;Here, the given length is 84.&amp;nbsp;Hence, the parallel vectors are&amp;nbsp;84v|v|&amp;nbsp;and&amp;nbsp;−84v|v|&amp;nbsp;Hence, the vector in the direction of v is&amp;nbsp;&amp;lt;−8442,−42042,33642&amp;gt;&amp;nbsp;and the vector in the opposite direction of v is&amp;nbsp;&amp;lt;8442,42042,−33642&amp;gt;.&amp;nbsp;The vector in direction of v is&amp;nbsp;&amp;lt;−8442,42042,−33642&amp;gt;&amp;nbsp;The vector in the opposite direction of v is&amp;nbsp;&amp;lt;8442,42042,−33642&amp;gt;.

karton · Accepted Answer

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→=Q→−P→=[025]−[171]=[−1−54]Next, we normalize d→ to obtain a unit vector d^:d^=d→‖d→‖=[−1−54](−1)2+(−5)2+42=[−16−5623]Finally, we can obtain two vectors parallel to PQ→ with length 84 by scaling d^:v1→=84·d^=84·[−16−5623]=[−14−7056]v2→=−84·d^=−84·[−16−5623]=[1470−56]Hence, two vectors parallel to PQ→ with a length of 84 are v1→=[−14−7056] and v2→=[1470−56].

user_27qwe · Accepted Answer

Step 1. Calculate the components of v→:v→=Q→−P→=⟨Qx−Px,Qy−Py,Qz−Pz⟩=⟨0−1,2−7,5−1⟩=⟨−1,−5,4⟩.Step 2. Normalize v→ to obtain a unit vector v^:‖v→‖=(−1)2+(−5)2+42=1+25+16=42.v^=v→‖v→‖=⟨−1,−5,4⟩42=⟨−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 · Accepted Answer

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→=Q→−P→=(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=42u→=d→‖d→‖=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))

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

Answered question

Answer & Explanation

New Questions in Precalculus