(1) find the projection of u onto v and (2) find the vector component of u orthogonal to v. u = ⟨6, 7⟩, v = ⟨1,4⟩

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Adnaan Franks · Accepted Answer

Step 1Recall that the projection of u onto v is given by:w1=projvu=(u⋅v||v||2)vGiven that u=&amp;lt;6,7&amp;gt; and v=&amp;lt;1,4&amp;gt;, we can find the projection of u onto v as shown below:w1=projvu=(u⋅v||v||2v=(&amp;lt;6,7&amp;gt;⋅&amp;lt;1,4&amp;gt;&amp;lt;1,4&amp;gt;⋅&amp;lt;1,)&amp;nbsp;=(6⋅1+7⋅41⋅1+4⋅4)&amp;lt;1,4&amp;gt;=3417&amp;lt;1,4&amp;gt;=&amp;lt;2,8&amp;gt;Step 2Remember that the following is the vector component of u that is orthogonal to v:Using the given vectors and the projection found in part (1), The vector component of u that is orthogonal to v can be found as shown below:w2=u−projvu=&amp;lt;6,7≻&amp;lt;2,8&amp;gt;=&amp;lt;(6−2),(7−8)&amp;gt;=&amp;lt;4,−1&amp;gt;1)&amp;nbsp;w1=&amp;lt;2,8&amp;gt;2)&amp;nbsp;w2=&amp;lt;4,−1&amp;gt;

Nick Camelot · Accepted Answer

Step 1: Calculate the dot product of 𝐮 and 𝐯:𝐮·𝐯=(6)(1)+(7)(4)=6+28=34Step 2: Calculate the magnitude of 𝐯:‖𝐯‖=(1)2+(4)2=1+16=17Step 3: Substitute the values into the projection formula:proj𝐯(𝐮)=(34172)⟨1,4⟩=(3417)⟨1,4⟩=⟨2,8⟩Therefore, the projection of 𝐮 onto 𝐯 is ⟨2,8⟩.To find the vector component of 𝐮 orthogonal to 𝐯, we can subtract the projection from 𝐮:comp𝐯(𝐮)=𝐮−proj𝐯(𝐮)Substituting the given values:comp𝐯(𝐮)=⟨6,7⟩−⟨2,8⟩=⟨4,−1⟩Hence, the vector component of 𝐮 orthogonal to 𝐯 is ⟨4,−1⟩.

Mr Solver · Accepted Answer

To find the projection of vector 𝐮 onto vector 𝐯 and the vector component of 𝐮 orthogonal to 𝐯, we can use the following formulas:(1) The projection of 𝐮 onto 𝐯, denoted as proj𝐯(𝐮), is given by:proj𝐯(𝐮)=𝐮·𝐯‖𝐯‖2·𝐯(2) The vector component of 𝐮 orthogonal to 𝐯, denoted as comp𝐯(𝐮), can be calculated as:comp𝐯(𝐮)=𝐮−proj𝐯(𝐮)Given 𝐮=⟨6,7⟩ and 𝐯=⟨1,4⟩, we can substitute these values into the formulas to find the solutions:(1) Projection of 𝐮 onto 𝐯:proj𝐯(𝐮)=⟨6,7⟩·⟨1,4⟩‖⟨1,4⟩‖2·⟨1,4⟩(2) Vector component of 𝐮 orthogonal to 𝐯:comp𝐯(𝐮)=⟨6,7⟩−proj𝐯(𝐮)

Eliza Beth13 · Accepted Answer

Step 1:1. The projection of 𝐮 onto 𝐯, denoted as proj𝐯𝐮, is given by:proj𝐯𝐮=(𝐮·𝐯‖𝐯‖2)𝐯2. The vector component of 𝐮 orthogonal to 𝐯, denoted as comp𝐯𝐮, can be obtained by subtracting the projection of 𝐮 onto 𝐯 from 𝐮:comp𝐯𝐮=𝐮−proj𝐯𝐮Step 2:Now let&#039;s calculate these values using the given vectors:Given:𝐮=⟨6,7⟩𝐯=⟨1,4⟩Step 3:1. To find the projection of 𝐮 onto 𝐯, we first calculate the dot product of 𝐮 and 𝐯:𝐮·𝐯=(6)(1)+(7)(4)=6+28=34Next, we find the squared magnitude of 𝐯:‖𝐯‖2=(1)2+(4)2=1+16=17Using these values, we can calculate the projection:proj𝐯𝐮=(3417)⟨1,4⟩Simplifying further:proj𝐯𝐮=(21)⟨1,4⟩=⟨2,8⟩Hence, the projection of 𝐮 onto 𝐯 is ⟨2,8⟩.Step 4:2. To find the vector component of 𝐮 orthogonal to 𝐯, we subtract the projection from 𝐮:comp𝐯𝐮=⟨6,7⟩−⟨2,8⟩Performing the subtraction:comp𝐯𝐮=⟨6−2,7−8⟩Simplifying further:comp𝐯𝐮=⟨4,−1⟩Therefore, the vector component of 𝐮 orthogonal to 𝐯 is ⟨4,−1⟩.

(a) find the projection of u onto v and (b) find the vector component of u ortho

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