Functions Involving Parameters, Vectors, and Matrices

The vector's direction changes, but its magnitude remains the same.

The vector becomes a zero vector.

Both the vector's magnitude and direction change.

The vector's magnitude changes, but its direction remains the same.

<-1, -1>

<0, 1>

<1, 0>

<0, 0>

An arrow drawn from the origin (0, 0) to the point (4, 3)

A dotted line drawn from the origin to the point (4, -3)

A solid line segment connecting points (0, 0) and (-4, -3)

An arc curving from the origin to the point (-4, 3)

An arc stretching from the origin to coordinates (3,-5)

An arrow starting at (0,0), moving right 3 units and down 5 units

Two separate arrows; one along x-axis for three units and one along y-axis for five units

An arrow starting at (3,-5), pointing towards the origin

One twelfth

One eighth

Two third

Four fifth

Third quadrant

Fourth quadrant

Second quadrant

First quadrant

The resulting vector points in exactly the opposite direction from before negation.

Magnitude remains unchanged while its projection onto any plane changes drastically due to inversion symmetry principles.

Only magnitude changes due to inherent asymmetries in spatial orientation under component-wise negation procedures.

There is no change to either magnitude or direction due to scalar cancellation properties.

<5, 7>

<9, 1>

<-9, -1>

<-5, -7>

$\sqrt{7}$

12

5

7

b=a/8 because direct proportionality would cause orthogonality without considering scale.

b=15/8*a because this represents another vector orthogonal to v.

b=-15*a because only adjusting scale makes it orthogonal to v.

b=-8/15*a because any scalar multiple could work but minimizing gives smallest non-zero values.