Algebra / English

Calculus 3 - Vector Projections & Orthogonal Components

The Organic Chemistry Tutor · YouTube · 12:14

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YouTube · The Organic Chemistry TutorOriginal video

IN THIS VIDEO

01Decompose relative to a vector
02Compute the projection
03Check the residual

SELECTED MOMENT

Decompose relative to a vector

SUGGESTED PREPARATION

Dot products and vector subtraction

READ & KEEP

The explanation, unpacked.

Reviewed learning material · Video analysis · English

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Compute the parallel projection and perpendicular residual of a vector in two and three dimensions.

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Chapters

1:30Decompose relative to a vector 4:30Compute the projection 6:00Check the residual

Learning script

Generated from the video's visuals and explanation; not verbatim speech.

For a nonzero reference vector v, split u into a component parallel to v and another perpendicular to v. These directions are relative to v, not necessarily the coordinate axes.

For $u=(3,5)$ and $v=(2,4)$ , the coefficient is ( $u\cdot v$ )/( $v\cdot v$ )= $26/20=13/10$ . The projection is ( $13/5$ , $26/5$ ).

Subtracting the projection gives ( $2/5$ ,− $1/5$ ). Its dot product with v is zero, confirming that the remaining component is orthogonal.

Knowledge cards

Decompose relative to a vector

For a nonzero reference vector v, split u into a component parallel to v and another perpendicular to v. These directions are relative to v, not necessarily the coordinate axes.

Compute the projection

For $u=(3,5)$ and $v=(2,4)$ , the coefficient is ( $u\cdot v$ )/( $v\cdot v$ )= $26/20=13/10$ . The projection is ( $13/5$ , $26/5$ ).

Check the residual

Subtracting the projection gives ( $2/5$ ,− $1/5$ ). Its dot product with v is zero, confirming that the remaining component is orthogonal.

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Orthogonal projections ExplanationAt 1:30
Why this connection?
For a nonzero reference vector v, split u into a component parallel to v and another perpendicular to v. These directions are relative to v, not necessarily the coordinate axes. For $u=(3,5)$ and $v=(2,4)$ , the coefficient is ( $u\cdot v$ )/( $v\cdot v$ )= $26/20=13/10$ . The projection is ( $13/5$ , $26/5$ ). Subtracting the projection gives ( $2/5$ ,− $1/5$ ). Its dot product with v is zero, confirming that the remaining component is orthogonal.

Calculus 3 - Vector Projections & Orthogonal Components

The explanation, unpacked.

Chapters

Learning script

Knowledge cards

Decompose relative to a vector

Compute the projection

Check the residual

Explore the knowledge in this video

Similar explanations

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