Find two unit vectors that make an angle of 60∘ with v=&lt;3,4&gt;

Question

Find two unit vectors that make an angle of 60∘ with v=&amp;lt;3,4&amp;gt;

AGRFTr · Accepted Answer

An angle of between two unit vectors is&amp;nbsp;60∘&amp;nbsp;with v=&amp;lt;3,4&amp;gt;&amp;nbsp;Considering that a= is the unit vector&amp;nbsp;|a|=x2+y2=1. Using our current dot producta⋅v=|a||v|cos⁡θ&amp;nbsp;&amp;lt;3,4&amp;gt;=1·32+42cos60°3x+4y=5(12)&amp;nbsp;6x+8y=5&amp;nbsp;y=18(5−6x)&amp;nbsp;In the unit vector length equation, replace with y.x2+y2=1&amp;nbsp;x2+(18(5−6x))2=1&amp;nbsp;x2+164(25−60x+36x2=64&amp;nbsp;100x2−60x−39=0&amp;nbsp;Time for a quadratic formulax=−b±b2−4ac2a&amp;nbsp;=−(60)±(−60)2−4(100)(−39)2(100)=60±19200200&amp;nbsp;=60±6400⋅3200=60±803200=3±4310&amp;nbsp;≈=−0.39282,0.99282&amp;nbsp;Find each&#039;s corresponding y.y≈18(5−6(−0.39282))≈0.919615&amp;nbsp;y≈18(5−6(0.99282))≈−0.119615&amp;nbsp;The two unit vectors are as follows:&amp;lt;−0.39282,0.919615&amp;gt;&amp;nbsp;and&amp;nbsp;&amp;lt;0.99282,−0.119615&amp;gt;

Don Sumner · Accepted Answer

Result:w1=⟨1,−110⟩andw2=⟨−1,1110⟩Solution:First, let&#039;s find the magnitude of 𝐯 using the formula:|𝐯|=32+42=5Next, we can find the unit vector in the direction of 𝐯 by dividing each component of 𝐯 by its magnitude:𝐮=⟨35,45⟩Now, we can find the angle between 𝐮 and the desired unit vectors. Since the angle between two vectors is given by the dot product formula:𝐮·𝐰=|𝐮||𝐰|cos(θ)where θ is the angle between 𝐮 and 𝐰.Since 𝐮 is a unit vector, its magnitude is 1, so we have:𝐮·𝐰=1·1·cos(60∘)=12Simplifying the equation, we have:35w1+45w2=12To find the two unit vectors 𝐰=⟨w1,w2⟩ that satisfy this equation, we can choose arbitrary values for one component and solve for the other component.Let&#039;s choose w1=1. Plugging this value into the equation, we have:35+45w2=12Solving for w2, we get:w2=−110Therefore, one unit vector that makes an angle of 60 degrees with 𝐯 is:w1=⟨1,−110⟩Now, let&#039;s choose w1=−1. Plugging this value into the equation, we have:−35+45w2=12Solving for w2, we get:w2=1110Therefore, another unit vector that makes an angle of 60 degrees with 𝐯 is:w2=⟨−1,1110⟩

RizerMix · Accepted Answer

To find two unit vectors that make an angle of 60 degrees with the vector 𝐯=⟨3,4⟩, we can follow these steps:First, let&#039;s normalize the vector 𝐯 to obtain a unit vector 𝐮 in the same direction. The formula to normalize a vector is given by:𝐮=𝐯‖𝐯‖where ‖𝐯‖ represents the magnitude (length) of vector 𝐯. In this case, we have:𝐯=⟨3,4⟩The magnitude of 𝐯 can be calculated using the formula:‖𝐯‖=v12+v22Substituting the values, we get:‖𝐯‖=32+42=9+16=25=5Now, we can calculate the unit vector 𝐮:𝐮=⟨3,4⟩5Simplifying, we have:𝐮=⟨35,45⟩So, the normalized unit vector 𝐮 is 𝐮=⟨35,45⟩.To find two unit vectors that make an angle of 60 degrees with 𝐯, we can use the rotation matrix. The rotation matrix is given by:𝐑=[cos(θ)−sin(θ)sin(θ)cos(θ)]where θ is the desired angle. In this case, θ=60∘.Substituting the values, we have:𝐑=[cos(60∘)−sin(60∘)sin(60∘)cos(60∘)]Calculating the values, we get:𝐑=[12−323212]To obtain the unit vectors, we multiply 𝐮 by 𝐑:𝐮1=𝐮·𝐑𝐮2=𝐮·𝐑Simplifying the calculations, we get:𝐮1=⟨35·12−45·32,35·32+45·12⟩𝐮2=⟨35·12+45·32,35·32−45·12⟩Simplifying further, we obtain:𝐮1=⟨3−2310,33+210⟩𝐮2=⟨3+2310,33−210⟩Therefore, the two unit vectors that make an angle of 60 degrees with 𝐯=⟨3,4⟩ are:𝐮1=⟨3−2310,33+210⟩𝐮2=⟨3+2310,33−210⟩

nick1337 · Accepted Answer

Step 1:First, we need to find the magnitude (length) of 𝐯. The magnitude of a vector 𝐯=(v1v2) is given by |𝐯|=v12+v22. In this case, |𝐯|=32+42=9+16=25=5.Next, we find the unit vector in the direction of 𝐯. The unit vector 𝐮 in the direction of 𝐯 is given by 𝐮=𝐯|𝐯|. Therefore, 𝐮=15(34)=(3545).Now, we can find two unit vectors that make an angle of 60 degrees with 𝐯 by using rotations. Let θ be the angle between 𝐯 and the desired unit vector. Since we want the angle to be 60 degrees, we have θ=60∘. To rotate a vector counterclockwise by an angle θ, we can use the rotation matrix R=(cos(θ)−sin(θ)sin(θ)cos(θ)).Step 2:Let 𝐮1 be the first unit vector that makes an angle of 60 degrees with 𝐯. We can find 𝐮1 by multiplying 𝐮 with the rotation matrix R:𝐮1=R𝐮=(cos(60∘)−sin(60∘)sin(60∘)cos(60∘))(3545)=(310−4103410+3103).Similarly, let 𝐮2 be the second unit vector that makes an angle of 60 degrees with 𝐯. We can find 𝐮2 by multiplying 𝐮 with the rotation matrix R rotated by 120 degrees:𝐮2=R(R𝐮)=(cos(60∘)−sin(60∘)sin(60∘)cos(60∘))(310−4103410+3103)=(−110−4103−310+2103).Therefore, the two unit vectors that make an angle of 60 degrees with 𝐯=(34) are:𝐮1=(310−4103410+3103)and𝐮2=(−110−4103−310+2103).

Find two unit vectors that make an angle of 60^\circ with v=<3,4>

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