Projection of u onto v and v onto u Given the vector u=&lt;−2,6,4&gt; and a vector v such that the vector projection of u onto v is &lt;2,4,4&gt;, and the vector projection of v onto u is &lt;−8,24,16&gt;. What is the vector v? Let v→=&lt;a,b,c&gt; Projection of v→ on→ u→ is given by: projuv=u→.v→|u|2u→=u→.v→(−2)2+62+42&lt;−2,6,4&gt; &lt;−8,24,16≥u→.v→(−2)2+62+42&lt;−2,6,4&gt; ⇒4&lt;−2,6,4≥u→.v→4+36+16&lt;−2,6,4&gt; ⇒&lt;−2,6,4≥u→.v→224&lt;−2,6,4&gt; On comparing u→.v→224=1 ⇒u→.v→=224 projvu=u→.v→|v|2v→=224|v|2&lt;a,b,c&gt; ⇒&lt;2,4,4≥224|v|2&lt;a,b,c&gt; Dividing both sides by 2 we get: ⇒&lt;1,2,2≥112|v|2&lt;a,b,c&gt;

Question

Projection of u onto v and v onto u  Given the vector u=&amp;lt;−2,6,4&amp;gt; and a vector v such that the vector projection of u onto v is &amp;lt;2,4,4&amp;gt;, and the vector projection of v onto u is &amp;lt;−8,24,16&amp;gt;. What is the vector v?  Let v→=&amp;lt;a,b,c&amp;gt;  Projection of v→ on→ u→ is given by:  projuv=u→.v→|u|2u→=u→.v→(−2)2+62+42&amp;lt;−2,6,4&amp;gt;  &amp;lt;−8,24,16≥u→.v→(−2)2+62+42&amp;lt;−2,6,4&amp;gt;  ⇒4&amp;lt;−2,6,4≥u→.v→4+36+16&amp;lt;−2,6,4&amp;gt;  ⇒&amp;lt;−2,6,4≥u→.v→224&amp;lt;−2,6,4&amp;gt;  On comparing u→.v→224=1  ⇒u→.v→=224  projvu=u→.v→|v|2v→=224|v|2&amp;lt;a,b,c&amp;gt;  ⇒&amp;lt;2,4,4≥224|v|2&amp;lt;a,b,c&amp;gt;  Dividing both sides by 2 we get:  ⇒&amp;lt;1,2,2≥112|v|2&amp;lt;a,b,c&amp;gt;

stomachdm · Accepted Answer

The projection of u onto v is a scalar times v. So from the given information we have  v=λ(2,4,4).   Hence projuv=v⋅uu⋅uu=λ914(−2,6,4).  It is given that this projection is (−8,24,16),so λ=569 and hence  v=569(2,4,4)

Karen Robbins · Accepted Answer

Here is a more tangent way to solve this problem:   Notice v and the projection of u onto v must have the same direction, therefore we can assume  v=λ(2,4,4),   in which λ is a constant to be determined. Now use the other condition to establish the equation  (−8,24,16)=(v,u)(u,u)u=36λ56(−2,6,4).  Solve this to get λ=569. Therefore, v=569(2,4,4).

Projection of u onto v and v onto u Given the vector u = <−2,6,

Answered question

Answer & Explanation

New Questions in Precalculus