Question

what compass heading represents 35° north of west?
?°
hint: compass headings are measured from north, going clockwise.

Explanation:

Step1: Understand Compass Heading

Compass headings start at north (0° or 360°) and measure clockwise. West is 270° from north (since north to east is 90°, east to south 90°, south to west 90°; 90×3 = 270°).

Step2: Calculate 35° North of West

North of west means we move 35° towards north from west. So we subtract 35° from west's heading (270°). Wait, no—wait, north is 0°, west is 270°. If we are 35° north of west, we are 35° towards north from west. So the angle from north clockwise: north to west is 270°, but moving 35° towards north from west means we take 270° - 35°? Wait, no, let's think again. North is 0°, east is 90°, south is 180°, west is 270°. If we are 35° north of west, that means from west, we turn 35° towards north. So the heading is 270° - 35° = 235°? Wait, no, wait: clockwise from north. So north (0°), then east (90°), south (180°), west (270°). If we are north of west, that's between north and west. So from north, going clockwise, west is 270°, but north of west is less than 270°. Wait, no—if you are 35° north of west, the angle from north clockwise is 180° + (90° - 35°)? Wait, no. Let's draw a mental map: north is up, west is left. 35° north of west means the direction is 35° towards north from west. So the angle from north, clockwise: north (0°), then west is 270°, but moving towards north from west (left) by 35°: so the heading is 270° - 35° = 235°? Wait, no, wait: 0° is north, 90° east, 180° south, 270° west. If you are 35° north of west, that is, the angle between the direction and west is 35° towards north. So the direction is west minus 35° (since north is towards decreasing the angle from west towards north). So 270° - 35° = 235°? Wait, let's check with another approach. The standard compass heading: north is 0°, each degree clockwise. So "north of west" means the angle between the direction and west is 35° towards north. So the heading is 180° + (90° - 35°)? Wait, no. Wait, west is 270°, north is 0°. The angle between north and west is 270°, but if we are 35° north of west, the angle from north clockwise is 270° - 35° = 235°? Wait, let's take an example: 45° north of west. Then it should be 225°? Wait, no, 45° north of west is northwest, which is 315°? Wait, no! Wait, I messed up. Oh no! Wait, northwest is 45° north of west, which is 315°? Wait, no, 45° north of west: from north, clockwise, northwest is 315°? Wait, no, north is 0°, east is 90°, south is 180°, west is 270°. Northwest is halfway between north and west, so 315°? Wait, 0° (north) to 90° (east) is 90°, 90° to 180° (south) 90°, 180° to 270° (west) 90°, 270° to 360° (north) 90°. So northwest is 315° (270° + 45°? No, 0° north, 90° east, 180° south, 270° west, 360° north. So northwest is 315°, which is 45° west of north, or 45° north of west. Ah! So 45° north of west is 315°. Oh! I see my mistake earlier. So north of west: the angle from north towards west. So 45° north of west is 315°, which is 360° - 45° = 315°, or 0° + 315°? Wait, no: from north, clockwise, 315° is towards west, 45° before west (since west is 270°? No, wait, no—315° is more than 270°, so that can't be. Wait, I'm confused. Let's use the correct method: compass heading is measured clockwise from north. So north is 0°, east is 90°, south is 180°, west is 270°. So "north of west" means the direction is between north and west, so the angle from north clockwise is 180° + (90° - θ), where θ is the angle north of west. Wait, no. Let's take 45° north of west: that's the same as 45° west of north. The angle from north, clockwise, for 45° west of north is 360° - 45° = 315°? Wait, no, 45° west of north: from north, turn 45° towards west (counterclockwise is west, but compass is clockwise). Wait, no—compass is clockwise from north. So north (0°), then east (90°) is clockwise, west is counterclockwise from north? No, no—compass heading is always clockwise from north. So 0°: north, 90°: east (clockwise), 180°: south, 270°: west (clockwise from north: north to east 90°, east to south 90°, south to west 90°, so 270° is west). So to get a direction that's north of west, we need to go from west (270°) towards north (0°), but since we measure clockwise from north, that direction is 270° - θ, where θ is the angle north of west. Wait, for 45° north of west: 270° - 45° = 225°? But 225° is south of west (since 180° is south, 225° is 45° south of west). Oh! Here's the mistake. I had the direction reversed. North of west is between north and west, so the angle from north clockwise should be more than 180°? Wait, no: north is 0°, east 90°, south 180°, west 270°. So between north (0°) and west (270°) clockwise is 270°, but that's the long way. The short way is counterclockwise, but compass heading is clockwise. Wait, no—compass heading is always clockwise from north, so all headings are 0° to 360°, clockwise. So north is 0°, east 90°, south 180°, west 270°. So a direction that's north of west is actually in the third quadrant? Wait, no: north is up, west is left, so north of west is up and left, which is the second quadrant (if we consider standard coordinate system with north as y-axis, east as x-axis). Wait, in standard coordinates, x is east, y is north. So west is -x, north is +y. So 35° north of west: the angle with the negative x-axis (west) is 35° towards positive y-axis (north). So in standard coordinates, that's 180° - 35° = 145° from the positive x-axis (east), but compass heading is from north (positive y-axis) clockwise. So to convert from standard coordinates (x east, y north) to compass heading (clockwise from y-axis): the compass heading θ is 90° - α, where α is the angle from positive y-axis (north) towards positive x-axis (east). Wait, no: if we have a direction with angle φ from positive y-axis (north) towards negative x-axis (west), then the compass heading is 360° - φ. Wait, let's take an example: 45° north of west. So from north (y-axis), we turn 45° towards west (negative x-axis). So the angle from north clockwise: north is 0°, east is 90°, south is 180°, west is 270°. So turning 45° towards west from north is 360° - 45° = 315°? But 315° is towards east-south? No, 315° is east of south? Wait, no, 315° clockwise from north: 0° north, 90° east, 180° south, 270° west, 315° is 45° east of south. Oh! I see the confusion. "North of west" is not the same as "west of north". Wait, "north of west" means the direction is closer to north than to west, while "west of north" is closer to west than to north. So 35° north of west: the angle between the direction and west is 35° towards north. So in terms of compass heading (clockwise from north): west is 270°, so moving towards north from west (i.e., decreasing the angle from west towards north) would be 270° - 35° = 235°? But 235° is south of west (since 180° is south, 235° is 55° south of west). That can't be. Wait, I think I have the direction of "north of west" wrong. Let's use the hint: compass headings are measured from north, going clockwise. So north (0°), then east (90°), south (180°), west (270°). So "north of west" means the direction is between north and west, so the angle from north clockwise is more than 180°? No, north is 0°, west is 270°, so between north (0°) and west (270°) clockwise is 270°, but that's the long way. The short way is counterclockwise, but compass is clockwise. Wait, no—compass heading is always clockwise, so all directions are 0° (north) to 360° (north again), clockwise. So "north of west" is a direction that is 35° towards north from west. So west is 270°, so towards north (which is 0°) from west (270°) is 270° - 35° = 235°? But 235° is 55° south of west (since 180° is south, 235° - 180° = 55° south of west). That's not north of west. I'm making a mistake here. Let's look up the correct method: To find the compass heading for θ° north of west: 1. West is 270° clockwise from north. 2. North of west means we are θ° towards north from west, so we subtract θ° from west's heading (270°). Wait, but that gives 270° - θ°, which is between 180° and 270°? No, 270° - 35° = 235°, which is between 180° (south) and 270° (west), so that's south of west. That's wrong. Ah! Here's the key: I had the direction reversed. "North of west" is actually west of north. Wait, no—terminology: "north of west" means the direction is northward from west, i.e., the angle between the direction and west is towards north. So in standard terms, if you face west, then turn 35° towards north, your direction is 35° north of west. In compass heading (clockwise from north), west is 270°, so turning towards north from west (which is counterclockwise from west) is equivalent to subtracting 35° from 270° (since compass is clockwise, moving towards north from west is decreasing the clockwise angle). But that gives 235°, which is south of west. That's a contradiction. Wait, no—let's use the correct formula: The compass heading for a direction that is α° north of west is 360° - α°. Wait, no, let's take α = 0°: 0° north of west is west, which is 270°, 360° - 0° = 360° = 0°, which is wrong. α = 90°: 90° north of west would be north, which is 0°, 360° - 90° = 270°, wrong. So that's not it. Wait, maybe the hint is wrong? No, the hint says compass headings are measured from north, going clockwise. So north (0°), east (90°), south (180°), west (270°). So "north of west" is a direction where the angle between the direction and west is towards north, so the direction is between north and west. So in terms of clockwise from north, north is 0°, west is 270°, so between them is 0° to 270° clockwise, but that's the long way. The short way is counterclockwise, but compass is clockwise. Wait, I think the mistake is in the interpretation: "35° north of west" means that the angle between the direction and the west axis is 35° towards the north axis. So in the coordinate system where north is y-axis, west is negative x-axis, the direction has an angle of 35° with the negative x-axis (west) towards the positive y-axis (north). So the angle with the positive y-axis (north) is 90° - 35° = 55° towards the negative x-axis (west). So the compass heading (clockwise from north) is 360° - 55° = 305°? Wait, 305° clockwise from north: 0° north, 90° east, 180° south, 270° west, 305° is 35° north of west (since from west (270°), moving 35° towards north (0°) clockwise? No, 305° - 270° = 35°, so from west (270°), moving 35° clockwise towards north (0°) would be 270° + 35° = 305°? Ah! There we go. I was subtracting before, but it's adding. Because clockwise from north: west is 270°, so moving towards north from west clockwise would be increasing the angle? No, clockwise from north: north (0°), east (90°), south (180°), west (270°), then back to north (360°). So from west (270°), moving clockwise towards north (360°) is increasing the angle. Wait, no—north is 0° and 360°, so west is 270°, and moving towards north from west clockwise is going from 270° to 360° (north). So 35° north of west clockwise would be 270° + 35° = 305°? Let's check with 45° north of west: 270° + 45° = 315°, which is northwest, which makes sense (45° north of west is northwest, and 315° clockwise from north is northwest). Yes! That's the correct way. So west is 270°, and north of west is towards north (clockwise from west towards north, which is 270° to 360°), so we add 35° to west's heading (270°). So 270° + 35° = 305°. Wait, let's verify: 305° clockwise from north: 0° north, 90° east, 180° south, 270° west, 305° is 35° past west towards north (since 305° - 270° = 35°), so that's 35° north of west. Yes, that makes sense. So the correct calculation is: west is 270° clockwise from north. North of west means moving towards north (clockwise) from west, so we add the angle (35°) to west's heading. So 270° + 35° = 305°.

Answer:

305