A set of three vectors {v1,v2,v3}{𝑣1,𝑣2,𝑣3}   in a vector space is an orthogonal set if each is vector is a non-zero vector, and if every vector is orthogonal with the other two. This means explicitly thatv1⋅v2=v2⋅v3=v3⋅v1=0.𝑣1⋅𝑣2=𝑣2⋅𝑣3=𝑣3⋅𝑣1=0.   So for example ⎧⎩⎨⎪⎪⎛⎝⎜341⎞⎠⎟,⎛⎝⎜5−2−7⎞⎠⎟,⎛⎝⎜1−11⎞⎠⎟⎫⎭⎬⎪⎪{(341),(5−2−7),(1−11)}   is an orthogonal set in R3𝑅3   . Can you find a set of three orthogonal vectors in R3𝑅3   which includes the vector ⎛⎝⎜4−12⎞⎠⎟?(4−12)?     Such a set is

Sure, we can find a set of three orthogonal vectors in R3 which includes the vector (4, -1, 2).

Orthogonal vectors are vectors that have a dot product of zero. This means that they are perpendicular to each other.

Let's denote our vectors as follows: v1 = (4, -1, 2) We need to find v2 and v3 such that they are orthogonal to v1 and to each other.

One simple way to find an orthogonal vector in R3 is to swap the first two elements of v1, change the sign of the new first element, and keep the third element as zero.

So, v2 can be (1, 4, 0).

To check if they are orthogonal, we calculate the dot product of v1 and v2: v1.v2 = 41 + -14 + 2*0 = 4 - 4 = 0

Now, to find v3, we can take the cross product of v1 and v2. The cross product of two vectors results in a vector that is orthogonal to both.

v3 = v1 x v2 = (-10 - 24, 21 - 04, 44 - -11) = (-8, 2, 17)

So, the set {(4, -1, 2), (1, 4, 0), (-8, 2, 17)} is a set of three orthogonal vectors in R3 which includes the vector (4, -1, 2).

Sure, we can find a set of three orthogonal vectors in R3 which includes the vector (4, -1, 2).

Orthogonal vectors are vectors that have a dot product of zero. This means that they are at right angles to each other.

Let's take the given vector v1 = (4, -1, 2). We need to find two more vectors v2 and v3 that are orthogonal to v1 and to each other.

Step 1: Find a second vector v2 that is orthogonal to v1.

We can do this by choosing two arbitrary values for the second vector and then solving for the third value using the condition of orthogonality (v1.v2 = 0).

Let's choose v2 = (a, b, 1) where a and b are any real numbers.

The dot product of v1 and v2 is:

v1.v2 = 4a - b + 2 = 0

Let's choose a = 1 and b = 4. So, v2 = (1, 4, 1) is orthogonal to v1.

Step 2: Find a third vector v3 that is orthogonal to both v1 and v2.

We can do this by taking the cross product of v1 and v2.

v3 = v1 x v2 = (4, -1, 2) x (1, 4, 1) = (-8, 0, 15)

So, the set of three orthogonal vectors in R3 which includes the vector (4, -1, 2) is {(4, -1, 2), (1, 4, 1), (-8, 0, 15)}.