What is the unit vector that is orthogonal to the plane containing # (20j +31k) # and # (32i-38j-12k) #?

Answer 1

The unit vector is #==1/1507.8<938,992,-640>#

The determinant is utilized to calculate the vector orthogonal to two vectros in a plane.

#| (veci,vecj,veck), (d,e,f), (g,h,i) | #
where #〈d,e,f〉# and #〈g,h,i〉# are the 2 vectors
Here, we have #veca=〈0,20,31〉# and #vecb=〈32,-38,-12〉#

Consequently,

#| (veci,vecj,veck), (0,20,31), (32,-38,-12) | #
#=veci| (20,31), (-38,-12) | -vecj| (0,31), (32,-12) | +veck| (0,20), (32,-38) | #
#=veci(20*-12+38*31)-vecj(0*-12-31*32)+veck(0*-38-32*20)#
#=〈938,992,-640〉=vecc#

Verification using the two dot method

#〈938,992,-640〉.〈0,20,31〉=938*0+992*20-640*31=0#
#〈938,992,-640〉.〈32,-38,-12〉=938*32-992*38+640*12=0#

So,

#vecc# is perpendicular to #veca# and #vecb#

The vector of units is

#hatc=vecc/||vecc||=(<938,992,-640>)/||<938,992,-640>||#
#=1/1507.8<938,992,-640>#
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Answer 2

To find the unit vector orthogonal to the plane containing the vectors (20j + 31k) and (32i - 38j - 12k), we first find the cross product of the two vectors and then normalize the result to obtain the unit vector. The cross product is calculated by taking the determinant of a 3x3 matrix formed by the basis vectors i, j, and k, with the components of the given vectors as coefficients. After calculating the cross product, normalize it by dividing each component by the magnitude of the resultant vector.The cross product of the given vectors is:

(20 * (-12) - 31 * 0) i + (-(32 * (-12)) - (20 * 0)) j + (32 * 31 - (-38 * 20)) k

Simplified, this becomes:

240i + 384j + 1040k

To find the unit vector, normalize this resultant vector:

Magnitude = sqrt(240^2 + 384^2 + 1040^2)

Unit vector = (240 / Magnitude)i + (384 / Magnitude)j + (1040 / Magnitude)k

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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