Projection: how much of one vector points along another

What to assume they know

• The scalar product, its definition and its component formula, from the previous topic
• The unit vector construction, from the second topic of the second module
• Scalar components and the three axis unit vectors, from the third topic of the second module
• Direction angles and direction cosines, from the second topic of the first module
• The cosine of an angle anywhere in a full turn, and where it is negative
• Absolute value of a real number

What they should be able to do

• Describe the projection vector of a vector on a directed line, in magnitude and in direction
• Say why the magnitude is written with modulus bars round the cosine, and where the sign goes instead
• Distinguish the projection vector from the projection, and say which of the two is a number
• Read the four panels of the chapter's projection figure and say what each angle range does to the direction
• Compute a projection with the unit vector formula, and recognise when it comes out negative
• Write the projection of one vector on another in the three equivalent forms the chapter gives
• State what happens at the two extreme angles and at the two right angles
• Compute a projection of one vector on another given in components
• Explain why a vector's three scalar components are the same three numbers as its projections along the axes
• Split a vector into a piece lying along a second vector and a piece square to it, and identify which of the two pieces is the projection vector
• Say what the chapter's Summary keeps of this section, and what it drops

Where it usually goes wrong

• "The projection of a vector is a vector." Sometimes. The chapter names the vector the projection vector and names its magnitude the projection. When a question asks for the projection it wants a number, and the number it wants is the signed one Observation 1 computes.
• "A projection cannot be negative." By the chapter's definition it is a magnitude, so it cannot; by the chapter's own formula three lines later it can, and does, whenever the angle is obtuse. The chapter uses one word for both. Compute the signed value and say what its sign means.
• "The angle in this section is the same angle as in the scalar product definition." It is measured through a full turn here and through half a turn there. Two of the chapter's four panels use ranges the scalar product definition does not admit.
• "The projection formula is a new formula to learn." It is one scalar product and one division. The three printed forms are one line rewritten twice.
• "To find a projection I need the angle." You need the scalar product and one magnitude. The angle never has to be computed, and computing it and then taking a cosine is the long way round.
• "Projecting onto a perpendicular vector is undefined." It is zero. Exercise 10.3 Q3 is exactly this case and the whole item collapses in one line.
• "Projecting onto a vector and projecting onto a line are different operations." They are the same operation; a vector supplies a direction, and the chapter's Observation 2 is Observation 1 with the unit vector spelled out.
• "The scalar components and the projections along the axes are two different triples." They are the same three numbers. That is the whole content of the Remark, and it is the only part of this section the Summary keeps.
• "The projection vector points along the original vector." It points along the line, either with it or against it. The original vector's direction is what is being measured, not what is being reported.

Questions to check understanding

• Describe the projection vector of a vector on a directed line, in magnitude and direction
• Compute the projection of one vector on another, both given in components — the form of Example 16 and Exercise 10.3 Q4
• Recognise a projection that vanishes without computing a magnitude — the form of Exercise 10.3 Q3
• Say what the projection vector is when the angle is zero, straight, or either right angle
• Write the projection of one vector on another in all three of the chapter's forms
• Explain why a projection may come out negative, and what the sign means
• Show that a vector's three scalar components are the same three numbers as its projections along the axes
• Derive a direction cosine as a scalar product against an axis unit vector
• Split a given vector into a piece along a second vector and a piece square to it, then check that the two close back onto the original and that the second piece's product with the first vector vanishes — the form of Miscellaneous Example 30

Examples worth working on the board

Values marked verified are worked out here from the chapter's own printed data; neither answers file was opened, and this chapter prints no answers to its exercises.

• The description that stands in for a definition (§10.6.2, Part II p. 357). A vector meets a directed line at some angle measured anticlockwise; the projection is then a vector along that line whose magnitude is the vector's magnitude times the modulus of the cosine, pointing the same way as the line or the opposite way according to whether the cosine is positive or negative. Verified as a description and not a numbered definition: the chapter's three numbered Definitions are on Part II pp. 339, 355 and 363, and this is not one of them. Section 1 should say so; a student who searches for "Definition" and finds nothing has not missed anything.
• The modulus bars round the cosine (§10.6.2, Part II p. 357). The magnitude is written with the cosine inside modulus bars. Verified as the same move the chapter made on Part II p. 346 when it put bars round the scalar in the scaling rule: a magnitude cannot be negative, so a quantity that may go negative has to be wrapped, and the information the bars discard is put back as a direction. Section 2 should draw the two rules side by side; it is the same idea twice and the second time is free.
• Two names, one word apart (§10.6.2, Part II p. 358). In one sentence the chapter names the vector the projection vector, and names its magnitude the projection. Verified as the chapter's own wording, and this is the topic's central hazard: one of the two is a vector, the other a number, and they differ by one word. Every later observation in the section uses one or the other and the reader has to keep track. Show both in section 3 and keep both labels visible for the rest of the explanation.
• Fig 10.20, all four panels (Part II p. 357). Read off the printed page: four small drawings of a horizontal directed line with a vector meeting it, labelled underneath with four angle ranges — first quadrant, second, third and fourth — and in each the projection drawn along the line with the foot of a dashed perpendicular marked. In the first the projection runs with the line; in the second it runs against it. Verified as a widening of the angle range: the scalar product's own Definition 2 on Part II p. 356 restricts its angle to half a turn, and this section measures through a full turn, using ranges beyond a straight angle in two of its four panels and again in Observation 4. The chapter never reconciles the two conventions, and Observation 2 then computes a projection with a scalar product, which is defined only on the narrower range. Section 4 should name the mismatch; see the note below.
• Observation 1, and the collision (§10.6.2, Part II p. 358). Take the unit vector along the line; then projecting anything onto that line is just taking the scalar product of the two. Verified as inconsistent with the definition three lines above it: that scalar product is negative whenever the angle is obtuse, while the projection was just defined as a magnitude and so cannot be. Both usages are the chapter's own, in the same section, on the same page. Section 5 should hold both in view and tell the student plainly which one an exam question will mean: when a question says "find the projection", it wants the signed number Observation 1 computes, because that is what every worked example and every exercise item in this chapter actually calculates.
• Observation 2, three forms (§10.6.2, Part II p. 358). The projection of one vector on another is written three ways: against the unit vector of the second; against the second divided by its own magnitude; and as the scalar product divided by the second magnitude. Verified as one formula written three ways: the second is the first with the hat expanded, and the third is the second with the division moved outside. Section 6 should show the three as one line rewritten twice, not as three facts.
• Observations 3 and 4 (§10.6.2, Part II p. 358). At a zero angle the projection vector is the original vector; at a straight angle it is the original reversed; and at either right angle it is the zero vector. Verified: these are the cosine's values at those angles fed through the description, and the reversal at a straight angle is exactly the case where the two meanings of projection come apart — the projection vector is the reversed vector, of positive magnitude, while Observation 1's formula returns the negative of that magnitude. Section 7 is thirty seconds and it is the cleanest demonstration of section 5's point.
• The Remark (§10.6.2, Part II p. 358). For a vector in component form, the three direction cosines are recomputed. Read off the printed page, because the text layer drops the entire display: the first cosine is written twice over, once as a scalar product against the first axis unit vector divided by the two magnitudes and then reduced to the first component over the magnitude; the second and third are printed only in reduced form. The Remark then says that the magnitude times each cosine is the projection along the corresponding axis, and concludes that the three scalar components are exactly the three projections. Verified: multiply the reduced form by the magnitude and the component reappears. Section 9 is the payoff of the topic and the bridge back to the direction cosine video, which carries the first half of this Remark; this topic carries the second half.
• The unit-vector corollary (§10.6.2, Part II p. 358). The Remark closes by saying that a unit vector may be written with its three direction cosines as its three components. Verified: it is the same sentence as Remark (iii) on Part II p. 349, arriving a second time from a different direction. Name it and hand it back to the direction cosine topic.
• Example 16 (Part II p. 359). Project a vector with components two, three and two onto a vector with components one, two and one. Verified: the scalar product is two plus six plus two, which is ten; the second magnitude is root six; so the projection is ten over root six, which the chapter tidies to five thirds of root six. The answer is positive, so the example does not expose the clash of section 5 — which is worth saying, because it means a student can complete this example without ever meeting the difficulty.
• Exercise 10.3 Q3 and Q4 (Part II p. 361). Project one two-component vector onto another; and project one three-component vector onto another. Verified: Q3's scalar product is one minus one, which is zero, so the projection is zero — the two vectors are perpendicular and the item is testing whether a student notices before computing a magnitude they will not need. Q4's scalar product is seven minus three plus fifty-six, which is sixty, and the second magnitude is root one hundred and fourteen, so the projection is sixty over that root, or equivalently ten root one hundred and fourteen over nineteen. These are the only two projection items in the chapter.
• Miscellaneous Example 30 (Part II p. 371). Two vectors are given in components, and the second is to be written as a sum of two pieces, one lying along the first and one square to it. Verified: writing the parallel piece as a multiple of the first vector and forcing the remainder's product with it to vanish gives a multiplier of one half, so the parallel piece has components three halves and minus one half, and the square piece has components one half, three halves and minus three. Their sum returns the original, and the second piece's product with the first vector is zero — both checked. This is the projection written as a decomposition, and it is the single most useful thing in the chapter for anything a student meets later: the parallel piece is exactly the projection vector this section defines, arrived at by a route the section never takes. The chapter buries it ten pages past §10.6.2, in the Miscellaneous Examples, and never connects the two — it does not use the word projection in the example, and §10.6.2 never mentions a decomposition. Section 10 exists to make the connection, and it is the strongest single reason this topic earns an explanation of its own rather than being folded into the scalar product. No other brief uses Example 30.

• What the Summary keeps (Part II p. 373). Verified as absent: of the eleven Summary bullets, none is about projection. The word occurs once in the whole Summary, inside the bullet that identifies the scalar components with the direction ratios, where it says those components are the projections along the axes. Confirmed by grep over the three Summary pages and on the page image of each. So the one sentence of §10.6.2 that survives revision is the Remark's conclusion, and not the section's own subject. Section 11 should say so and hand the student the two exercise items as the whole of what is examinable from the printed book.

Figures to have open

• A redraw of Fig 10.20 (Part II p. 357) as four panels for sections 1, 4 and 7: a horizontal directed line, a vector meeting it, the dashed perpendicular and its foot, and the projection drawn along the line, with the four angle ranges labelled underneath as the chapter labels them. All four panels must be drawn at one size with one type size, because they are peers and the chapter prints them as a set.
• A single reusable frame for sections 1, 2 and 5, in which one vector swings against a fixed line while the projection follows it, so that the magnitude rule, the sign question and the collision are all made on one drawing.
• A three-axis frame for section 9 with one vector and its three components, each component able to light independently and carry two labels at once — its component name and its projection name. Not in the book; the chapter prints no figure with the Remark.
• A drawing for section 10, entirely not in the book: the two vectors of Miscellaneous Example 30 from a common tail, the second one then splitting into an arrow along the first and an arrow square to it, a right-angle mark at the join, and the two closing back onto the original. The chapter prints no figure with Example 30 — checked on the page image of Part II p. 371, which carries the worked solution and nothing drawn — which is exactly why the link back to Fig 10.20 is invisible on the page.
• No figure is needed for sections 3, 6, 8 or 11 beyond the kit shapes named above. §10.6.2 prints exactly one figure, and it is Fig 10.20; confirmed on the page images of Part II pp. 357 and 358.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 10 "Vector Algebra", §10.6.2 Projection of a vector on a line, Part II pp. 357–358, with Fig 10.20 panels (i) to (iv)
• The naming of the projection vector and the projection, and Observations 1 to 4, Part II p. 358
• The Remark on direction cosines and projections along the axes, Part II p. 358
• Definition 2 and its angle range, §10.6.1, Part II p. 356, cited for the mismatch
• Example 16, Part II p. 359; Miscellaneous Example 30, Part II p. 371
• Exercise 10.3, questions 3 and 4, Part II p. 361
• Remark (iii), §10.5.1, Part II p. 349, cited for the repeated corollary
• Summary, the scalar components bullet, Part II p. 373, cited for what it keeps and what it drops