Vectors: Test Review

Key facts

Where marks get lost

• ×Treating speed and velocity (or distance and displacement) as the same quantity. Speed is a scalar; velocity is speed together with a direction. (Vectors, Scalars and Direction)
• ×Reading N $40°$ W as $40°$ from west. The first letter is the starting direction, so it is $40°$ from north, toward west. (Vectors, Scalars and Direction)
• ×Measuring a standard angle clockwise, or a true bearing counterclockwise. Standard angles turn counterclockwise from the positive $x$-axis; bearings turn clockwise from north. (Vectors, Scalars and Direction)
• ×Writing $\overrightarrow{AB}=\overrightarrow{CD}$ for opposite sides of parallelogram $ABCD$. The sides are parallel and equal in length, but the arrows point opposite ways, so $\overrightarrow{CD}=-\overrightarrow{AB}$. (Vectors, Scalars and Direction)
• ×Thinking a negative scalar such as $-5$ °C has a direction. The sign places it on a scale; it is still a scalar. (Vectors, Scalars and Direction)
• ×Adding magnitudes: $|\vec{a}+\vec{b}|=|\vec{a}|+|\vec{b}|$ only when the vectors point the same way (or one of them is $\vec{0}$). In general $|\vec{a}+\vec{b}|\le |\vec{a}|+|\vec{b}|$. (Adding, Subtracting and Scaling Vectors)
• ×Using the $60°$ angle between the vectors directly in the cosine law for $\vec{a}+\vec{b}$. In the tip-to-tail triangle the interior angle is $180°-60°=120°$. (Adding, Subtracting and Scaling Vectors)
• ×Drawing $\vec{a}-\vec{b}$ from the head of $\vec{a}$ to the head of $\vec{b}$. That arrow is $\vec{b}-\vec{a}$; with tails together, $\vec{a}-\vec{b}$ points TO the head of $\vec{a}$. (Adding, Subtracting and Scaling Vectors)
• ×Forgetting that a negative scalar reverses the direction: $-2\vec{a}$ is twice as long as $\vec{a}$ AND points the opposite way. (Adding, Subtracting and Scaling Vectors)
• ×Joining tail to tail when using the triangle law. The triangle law needs head to tail; tail to tail is the parallelogram law, where the sum is the diagonal. (Adding, Subtracting and Scaling Vectors)
• ×Trusting ${\tan }^{-1}(y/x)$ without checking the quadrant. $(-5,12)$ and $(5,-12)$ give the same calculator output, but their directions differ by $180°$. (Cartesian Vectors in 2-Space and 3-Space)
• ×Subtracting tail minus head for $\overrightarrow{AB}$. The vector from $A$ to $B$ is $B-A$; doing $A-B$ gives the opposite vector. (Cartesian Vectors in 2-Space and 3-Space)
• ×Using a bearing as if it were a standard angle in $x=r\cos \theta$. A true bearing of $120°$ is a standard angle of $90°-120°+360°=330°$, not $120°$. (Cartesian Vectors in 2-Space and 3-Space)
• ×Squaring a negative component wrongly, for example typing $-5^2$ (which a calculator reads as $-25$). In fact ${(-5)}^2=25$, so a negative component adds to the magnitude just as a positive one does. (Cartesian Vectors in 2-Space and 3-Space)
• ×Mixing up the coordinate planes: points on the $xz$-plane have $y=0$, not $z=0$. (Cartesian Vectors in 2-Space and 3-Space)
• ×Drawing the wind the wrong way. A wind FROM the south points north; drawing it pointing south turns a tailwind into a headwind. (Applications: Forces, Wind and Current)
• ×Using cosine for the east component of a bearing. For an angle measured from NORTH, east $=v\sin \beta$ and north $=v\cos \beta$, the reverse of the standard-angle formulas. (Applications: Forces, Wind and Current)
• ×Using the angle between the vectors (tail to tail) as the interior angle of the tip-to-tail triangle. For the plane, the triangle angle is $180°-27°=153°$, not $27°$. (Applications: Forces, Wind and Current)
• ×Dividing the river width by the ground speed to find the crossing time. Only the component straight across the river counts. (Applications: Forces, Wind and Current)
• ×Giving the ground speed without the track, or the equilibrant with the resultant's direction. A vector answer needs magnitude and direction. (Applications: Forces, Wind and Current)
• ×Writing the dot product as a vector, for example $(2,-4,6)$ instead of $2-4+6=4$. Multiply matching components, then ADD them. (The Dot Product)
• ×Dividing by the wrong magnitude in a projection. The projection of $\vec{a}$ onto $\vec{b}$ divides by $|\vec{b}|$ (the vector you project ONTO). (The Dot Product)
• ×Dividing by $|\vec{b}|$ instead of $|\vec{b}{|}^2$ in the vector projection, which gives a vector of the wrong length. (The Dot Product)
• ×Measuring $\theta$ with the vectors placed tip to tail. The angle in $|\vec{a}||\vec{b}|\cos \theta$ is between the vectors drawn tail to tail. (The Dot Product)
• ×Assuming the dot product is associative. $(\vec{a}\cdot \vec{b})\cdot \vec{c}$ is meaningless, because $\vec{a}\cdot \vec{b}$ is a number. (The Dot Product)
• ×Getting the middle component backwards. It is $a_3{b}_1-a_1{b}_3$; writing $a_1{b}_3-a_3{b}_1$ flips its sign. Dotting the answer with $\vec{a}$ and $\vec{b}$ catches this immediately. (The Cross Product)
• ×Assuming $\vec{a}\times \vec{b}=\vec{b}\times \vec{a}$. Swapping the order reverses the direction. (The Cross Product)
• ×Forgetting the $\frac{1}{2}$ for a triangle, or using $\cos \theta$ in the magnitude formula. Cross products use $\sin \theta$; dot products use $\cos \theta$. (The Cross Product)
• ×Trying to take a cross product of 2-D vectors. The cross product is defined in 3-space; write $(x,y)$ as $(x,y,0)$ if you need it. (The Cross Product)
• ×Leaving a negative volume. The scalar triple product can be negative (it depends on the order of the vectors), so take its absolute value. (The Cross Product)

Practice test

24 questions from across the unit. Each one shows the lesson it comes from.

Endless drill

Fresh numbers every time, for the calculation questions in this unit.