PMDC Unit 37 · 3 outcomes

Vectors and Equilibrium

A vector has size and direction. 5 m east is a vector. 5 m by itself is a scalar. Unit 1 on the PMDC list is only three skills: add vectors with perpendicular parts, the dot product, and the cross product.

Open the Unit 37 quiz Worked numericals

1.1

Addition of vectors (rectangular components)

PMDC: Determine the sum of vectors using perpendicular components.

Any vector A sitting at an angle $θ$ to the x-axis can be split into two legs of a right triangle. The x-leg is $A_{x}$ . The y-leg is $A_{y}$ . Those two legs are the rectangular components.

To add several vectors, never chase arrows around the page. Break every vector into $A_{x}$ and $A_{y}$ . Add all the x-parts. Add all the y-parts. The totals $R_{x}$ and $R_{y}$ are the legs of the resultant. Pythagoras gives the size. The inverse tangent gives the angle.

A component is negative if it points left (x) or down (y). Keep the sign. That is the whole method.

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A_{x} = 10 cos 6 0^{\circ} = 5.0 A_{y} = 8.7

$A_{x} = A cos θ$
$A_{y} = A sin θ$
$R_{x} = Σ A_{x} R_{y} = Σ A_{y}$
$R = R_{x}^{2} + R_{y}^{2}$
$θ$ = tan⁻¹ ( $R_{y} / R_{x}$ ) (watch the quadrant)

Hold these

Components add. Magnitudes do not, unless the vectors are in the same line.
$θ$ is measured from the +x axis unless the stem says otherwise.
If two vectors are perpendicular, $R = \sqrt (A^{2}$ + B²).

1.2

Scalar product (dot product)

PMDC: Describe the scalar product of two vectors in terms of the angle between them.

The scalar product (dot product) of A and B is a number, not a new arrow. It tells you how much one vector lies along the other.

If the vectors are parallel, $θ = 0^{\circ}$ , cos $0 = 1$ , and A · B is as large as it can be: AB. If they are perpendicular, $θ = 9 0^{\circ}$ , cos $9 0^{\circ}$ = 0, so the dot product is zero. Work $W = F$ · s is a scalar product — that is why only the part of F along the displacement counts.

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A \cdot B = 4 \times 5 \times cos 6 0^{\circ} = 10.0

$A \cdot B = A B cos θ$
$A \cdot B = A_{x} B_{x} + A_{y} B_{y} + A_{z} B_{z}$
$^\cdot ^= ^\cdot ^= k^\cdot k^= 1$
$^\cdot ^= ^\cdot k^= k^\cdot ^= 0$

Hold these

Dot product is a scalar. No direction.
A · $B = 0$ means A and B are perpendicular (if neither is zero).
$A \cdot A = A^{2} .$

1.3

Vector product (cross product)

PMDC: Describe the vector product of two vectors in terms of the angle between them.

The vector product (cross product) of A and B is a new vector. Its size is AB sin $θ$ . Its direction is perpendicular to both A and B — use the right-hand rule. Fingers from A to B, thumb points along A × B.

If the vectors are parallel, $θ = 0^{\circ}$ , sin $0 = 0$ , so the cross product is zero. Torque $τ$ = r × F and the magnetic force $F = q$ v × B are cross products. That is why they are largest at $9 0^{\circ}$ and zero when the two arrows line up.

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∣ A \times B ∣ = 3 \times 4 \times sin 9 0^{\circ} = 12

$∣ A \times B ∣ = A B sin θ$
direction of A × B: right-hand rule, ⊥ to both A and B
$^\times ^= k^^\times k^= ^k^\times ^= ^$
$^\times ^= ^\times ^= k^\times k^= 0$
A × $B = -$ B × A (order matters)

Hold these

Cross product is a vector. It stands out of the plane of A and B.
A × $B = 0$ means A and B are parallel (or one is zero).
Swapping the order flips the direction.

Unit 37 formula sheet

Write these until they are cheaper than thinking, then open the quiz.

$A_{x} = A cos θ$
$A_{y} = A sin θ$
$R_{x} = Σ A_{x} R_{y} = Σ A_{y}$
$R = R_{x}^{2} + R_{y}^{2}$
$θ$ = tan⁻¹ ( $R_{y} / R_{x}$ ) (watch the quadrant)
$A \cdot B = A B cos θ$
$A \cdot B = A_{x} B_{x} + A_{y} B_{y} + A_{z} B_{z}$
$^\cdot ^= ^\cdot ^= k^\cdot k^= 1$
$^\cdot ^= ^\cdot k^= k^\cdot ^= 0$
$∣ A \times B ∣ = A B sin θ$
direction of A × B: right-hand rule, ⊥ to both A and B
$^\times ^= k^^\times k^= ^k^\times ^= ^$
$^\times ^= ^\times ^= k^\times k^= 0$
A × $B = -$ B × A (order matters)

Open the Unit 37 quiz