Determine all vectors vv that are orthogonal to u. u=(4,-1,0)

Question

Determine all vectors vv that are orthogonal to u.  u=(4,-1,0)

rerCessbalmuh · Accepted Answer

If u*v=0, then vectors u and v are orthogonal.
We have to find all vectors

v = (v_{1}, v_{2}, v_{3}) \in R^{3}

such that u*v=0.

(4, - 1, 0) \cdot (v_{1}, v_{2}, v_{3}) = 0

4 v_{1} - v_{2} = 0

4 v_{1} = v_{2}

v_{1} = \frac{v_{2}}{4}

So, all vectors

v = (\frac{v_{2}}{4}, v_{2}, v_{3}), w h e r e v_{2}, v_{3}

are any numbers are orthogonal to u.
Result:

v = (\frac{v_{2}}{4}, v_{2}, v_{3})

star233 · Accepted Answer

Let v = (x, y, z) be a vector orthogonal to u. Then, we have:

u \cdot v = (4, - 1, 0) \cdot (x, y, z) = 4 x + (- 1) y + 0 z = 4 x - y = 0

.
From this equation, we can solve for y in terms of x:
-4x + y = 0
y = 4x.
Therefore, any vector v orthogonal to u can be represented as v = (x, 4x, z), where x and z are arbitrary real numbers.

karton · Accepted Answer

Answer:𝐯=(v1,4v1,v3) where v1 and v3 can be any real numbers.Explanation:𝐮·𝐯=0Substituting the values of 𝐮, we have:(4,−1,0)·(v1,v2,v3)=0This gives us the following equation:4v1−v2=0To find the solutions, we can express v2 in terms of v1:v2=4v1Therefore, the vectors orthogonal to 𝐮 can be represented as:𝐯=(v1,4v1,v3) where v1 and v3 can be any real numbers.

user_27qwe · Accepted Answer

Step 1:The dot product of two vectors a = (a1, a2, a3) and b = (b1, b2, b3) is given by:a·b=a1b1+a2b2+a3b3.In this case, we have:u·v=(4)(v1)+(−1)(v2)+(0)(v3)=4v1−v2=0.Rearranging the equation, we have:4v1=v2.This equation tells us that any vector v = (v1, v2, v3) satisfying the condition 4v1=v2 will be orthogonal to u.Step 2:Expressing the solution in vector form, we can write v = (v1, 4v1, v3), where v1 and v3 can be any real numbers.Hence, the set of all vectors orthogonal to u is given by:𝐯={(v1,4v1,v3):v1,v3∈ℝ}.

Determine all vectors vv that are orthogonal to u. u=(4,-1,0)

Answered question

Answer & Explanation

New Questions in Linear algebra