What's the visual and mathematical difference between a vector projection of #a# onto #b# and an orthogonal projection of #a# onto #b#? Are they just different ways to say the same thing?
Despite that the magnitude and direction are the same, there is a nuance. The orthogonalprojection vector is on the line in which the other vector is acting. The other could be parallel
Vector projection is just projection in the direction of the other vector.
In direction and magnitude, both are the same. Yet, the orthogonalprojection vector is deemed to be on the line in which the other vector is acting. Vector projection may possibly be parallel
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The visual and mathematical difference between a vector projection of (a) onto (b) and an orthogonal projection of (a) onto (b) lies in their outcomes and interpretations.

Vector Projection of (a) onto (b):
 The vector projection of (a) onto (b) results in a vector that is parallel to (b) and is a scaled version of (b).
 Mathematically, the vector projection of (a) onto (b) is given by:
(\text{proj}_b(a) = \frac{a \cdot b}{\lVert b \rVert^2} \cdot b)  This projection represents the component of vector (a) that lies in the direction of vector (b).

Orthogonal Projection of (a) onto (b):
 The orthogonal projection of (a) onto (b) results in a vector that is perpendicular to (b) and lies in the plane spanned by (b).
 Mathematically, the orthogonal projection of (a) onto (b) is given by:
(\text{proj}_{b^\perp}(a) = a  \text{proj}_b(a))  This projection represents the component of vector (a) that is perpendicular to vector (b).
They are not just different ways to say the same thing. While both projections involve projecting vector (a) onto the direction of vector (b), the outcomes differ in terms of direction and magnitude. The vector projection of (a) onto (b) results in a vector parallel to (b), while the orthogonal projection results in a vector perpendicular to (b).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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