Vectors

Explanation

• A vector records magnitude and direction; in two or three dimensions it can be written in component form or with the unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$.
• Work component by component, keeping the $x$, $y$ and $z$ entries aligned; subtract the initial point from the final point to form a displacement vector.
• For example, from $A(1,2,3)$ to $B(4,0,5)$ the displacement is $\overrightarrow{AB}=(3,-2,2)$.
• A vector has no fixed location, whereas a point does; a common error is to confuse the coordinates of an endpoint with the components of the displacement leading to it.

Explanation

• For $\mathbf{v}=(a,b)$, its magnitude is $|\mathbf{v}|=\sqrt{a^2+b^2}$ and a direction angle must be stated relative to a specified axis or bearing convention.
• A vector of magnitude $r$ at angle $\theta$ anticlockwise from the positive $x$-axis has components $(r\cos\theta,r\sin\theta)$; use signs or a quadrant-aware angle calculation when reversing the process.
• For example, magnitude $10$ at $30^\circ$ above the positive $x$-axis gives $(10\cos30^\circ,10\sin30^\circ)=(5\sqrt3,5)$.
• The value from $\tan^{-1}(b/a)$ alone can select the wrong quadrant; a common error is to report an acute reference angle without checking the component signs.
• A unit vector in the direction of non-zero $\mathbf a$ is $\mathbf a/|\mathbf a|$; divide every component by the vector's magnitude.

Explanation

• Vector addition combines successive displacements: placing the tail of $\mathbf{b}$ at the head of $\mathbf{a}$ makes the resultant from the first tail to the final head equal to $\mathbf{a}+\mathbf{b}$.
• Add corresponding components and multiply every component by a scalar; subtraction is addition of the opposite vector.
• The vector $k\mathbf{a}$ is parallel to $\mathbf{a}$, has magnitude $|k||\mathbf{a}|$, and points in the reverse direction when $k<0$.
• A common error is to multiply only one component by a scalar or to draw vectors head-to-head instead of using a head-to-tail or parallelogram construction.

Explanation

• The position vector of a point $A$ is $\overrightarrow{OA}$ from a fixed origin $O$; if these vectors are $\mathbf{a}$ and $\mathbf{b}$, then $\overrightarrow{AB}=\mathbf{b}-\mathbf{a}$.
• Find a displacement by subtracting position vectors in final-minus-initial order, then find distance by taking the magnitude of that displacement.
• A point dividing $AB$ internally in the fraction $t$ from $A$ to $B$ has position vector $\mathbf{a}+t(\mathbf{b}-\mathbf{a})$.
• Distance is a non-negative scalar, not a vector; a common error is to quote $\mathbf{b}-\mathbf{a}$ as the distance without calculating its magnitude.

Explanation

• Vector models turn geometrical displacements or forces into component equations; equilibrium means that the vector sum of all forces is zero.
• Choose and state positive coordinate directions, resolve every vector consistently, then equate components or use position-vector relationships.
• In a parallelogram with adjacent position vectors $\mathbf{a}$ and $\mathbf{b}$, the opposite vertex has position vector $\mathbf{a}+\mathbf{b}$ and both diagonals share midpoint $\frac12(\mathbf{a}+\mathbf{b})$.
• A common error is to balance force magnitudes without balancing directions; equal numerical magnitudes do not guarantee equilibrium unless the vector sum is zero.